GCSE Maths: The probability scale and probability of single events (Foundation)
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The probability scale and probability of single events
We use chance every day: “It will probably rain”, “I might win the raffle”, “There is no way I will finish that today.” In maths we measure chance with a number called a probability. The first words to learn are these. An experiment is anything we do where the result is not certain, such as rolling a die. Each possible result is an outcome. An event is the outcome (or group of outcomes) we are interested in, such as “the die shows an even number”. If every outcome has the same chance, we say the outcomes are equally likely, and a fair experiment, where nothing is rigged, is called random. A coin that is not fair is called biased.
The probability scale. Every probability is a number from 0 to 1. An event that cannot happen has probability 0 (impossible). An event that must happen has probability 1 (certain). An event that is as likely to happen as not has probability 1/2 (an even chance). Everything else lies between: the nearer to 1, the more likely. Probabilities can be written as fractions, decimals or percentages, so 1/2 = 0.5 = 50%. They are never written as ratios, and they are never bigger than 1 (or 100%) or smaller than 0.
Theoretical probability. When all the outcomes are equally likely we can calculate the probability without doing the experiment. Picture a bag with 3 red counters and 2 blue counters. There are 5 counters and each one is equally likely to be picked. Three of them are red, so the chance of red is 3 out of 5.
The rule is: probability = number of favourable outcomes ÷ total number of outcomes. “Favourable” just means “the ones we are looking for”. We write P(red) for “the probability of red”. To use the rule: (1) list or count all the possible outcomes, this is the bottom of the fraction; (2) count the outcomes that match the event, this is the top; (3) simplify the fraction. A single die has six equally likely outcomes, 1, 2, 3, 4, 5, 6. Three of them are even (2, 4, 6), so P(even) = 3/6 = 1/2.
Key ideas
- Probability is a number from 0 (impossible) to 1 (certain). 1/2 is an even chance.
- Write probabilities as fractions, decimals or percentages, never as a ratio.
- For equally likely outcomes: probability = favourable outcomes ÷ total outcomes.
- The bottom of the fraction is all the outcomes. The top is only the ones we want.
- Simplify fractions: 3/6 = 1/2. Probability can never be more than 1 or less than 0.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6
Total number of outcomes = 6 - Pick out the outcomes where the score is a prime number. These are the favourable outcomes. Their number goes on the top.2, 3, 5
Number of favourable outcomes = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the score is a prime number) = 36
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 3. The value does not change but the numbers are smaller.3 ÷ 3 = 1 and 6 ÷ 3 = 2
36 = 12
Check. The other 3 outcomes do not fit the event, so P(the event does not happen) = 12. Then 36 + 36 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of counters. Every counter is equally likely to be picked, so add up all the counters. This is the bottom of the fraction.5 + 3 + 2 = 10
- Count the counters that are red and green. These are the favourable outcomes. This number goes on the top.5 + 2 = 7
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the counter is red or green) = 710
Check. The other 3 outcomes do not fit the event, so P(the event does not happen) = 310. Then 710 + 310 = 1010 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction. Count every section, even when a number appears more than once.1, 2, 3, 3, 4, 5, 5, 5
Total number of sections = 8 - Pick out the outcomes where the spinner lands on 5. These are the favourable outcomes. Their number goes on the top.5, 5, 5
Number of favourable sections = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the spinner lands on 5) = 38
Check. The other 5 outcomes do not fit the event, so P(the event does not happen) = 58. Then 38 + 58 = 88 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Method. Probabilities can be written as fractions, decimals or percentages. Change the fraction by making the denominator 100 or by dividing.
- Start with the fraction. A fraction, a decimal and a percentage are three ways of writing the same number.720
- Make the denominator 100. A percentage is a number out of 100. Multiply the top and the bottom by the same number.20 × 5 = 100, so 7 × 5 = 35
720 = 35100 - Write it as a decimal and a percentage. 35 out of 100 is 35%, and 35 hundredths is 0.35.35100 = 0.35 = 35%
Check. Multiply the decimal by the denominator: 0.35 × 20 = 7, which is the numerator 7. ✓
Watch out. A probability as a percentage can be written 25%, as a decimal 0.25 and as a fraction 14. They are all the same number.
Watch out
The bottom of the fraction is all the outcomes, not the number of outcomes you do not want. In a bag with 3 red and 2 blue counters, P(red) is 3/5, not 3/2 and not 3 : 2. And a number that appears twice on a spinner counts twice.Exercise 23A The probability scale and single events
Non-calculatorMethod. Work out (or reason out) the probability, then place it on the probability scale from 0 to 1.
- Work out how likely the event is. We need a number to compare with the scale.A coin has 2 equally likely sides and 1 is heads, so P(heads) = 1/2.
- Compare with the probability scale. The scale runs from impossible (0) through even chance (1/2) to certain (1).12: The probability is exactly 1/2, as likely as not.
Check. The complement has probability 12, and the two add to 1. ✓
Watch out. “Unlikely” does not mean “impossible”. Only a probability of exactly 0 is impossible, and only exactly 1 is certain.
Method. Work out (or reason out) the probability, then place it on the probability scale from 0 to 1.
- Work out how likely the event is. We need a number to compare with the scale.Friday always follows Thursday, so this must happen: probability 1.
- Compare with the probability scale. The scale runs from impossible (0) through even chance (1/2) to certain (1).1: The probability is 1, so it must happen.
Check. The complement has probability 0, and the two add to 1. ✓
Watch out. “Unlikely” does not mean “impossible”. Only a probability of exactly 0 is impossible, and only exactly 1 is certain.
Method. Work out (or reason out) the probability, then place it on the probability scale from 0 to 1.
- Work out how likely the event is. We need a number to compare with the scale.A die only has the numbers 1 to 6, so a 7 cannot happen: probability 0.
- Compare with the probability scale. The scale runs from impossible (0) through even chance (1/2) to certain (1).0: The probability is 0, so it cannot happen.
Check. The complement has probability 1, and the two add to 1. ✓
Watch out. “Unlikely” does not mean “impossible”. Only a probability of exactly 0 is impossible, and only exactly 1 is certain.
Method. Work out (or reason out) the probability, then place it on the probability scale from 0 to 1.
- Work out how likely the event is. We need a number to compare with the scale.P(red) = 1 out of 10 counters = 1/10.
- Compare with the probability scale. The scale runs from impossible (0) through even chance (1/2) to certain (1).110: 110 is less than 1/2, so it is more likely not to happen.
Check. The complement has probability 910, and the two add to 1. ✓
Watch out. “Unlikely” does not mean “impossible”. Only a probability of exactly 0 is impossible, and only exactly 1 is certain.
Method. Work out (or reason out) the probability, then place it on the probability scale from 0 to 1.
- Work out how likely the event is. We need a number to compare with the scale.P(red) = 9 out of 10 counters = 9/10.
- Compare with the probability scale. The scale runs from impossible (0) through even chance (1/2) to certain (1).910: 910 is more than 1/2, so it is more likely to happen than not.
Check. The complement has probability 110, and the two add to 1. ✓
Watch out. “Unlikely” does not mean “impossible”. Only a probability of exactly 0 is impossible, and only exactly 1 is certain.
Method. Work out (or reason out) the probability, then place it on the probability scale from 0 to 1.
- Work out how likely the event is. We need a number to compare with the scale.The outcomes 1 and 2 match, so P = 2 out of 6 = 1/3.
- Compare with the probability scale. The scale runs from impossible (0) through even chance (1/2) to certain (1).13: 13 is less than 1/2, so it is more likely not to happen.
Check. The complement has probability 23, and the two add to 1. ✓
Watch out. “Unlikely” does not mean “impossible”. Only a probability of exactly 0 is impossible, and only exactly 1 is certain.
Method. Impossible means it can never happen, which is the bottom of the scale.
- Recall the end points of the scale. The scale runs from 0 to 1.0 = impossible, 1 = certain
- Choose the one for “cannot happen”. Probability = 0 (or 0%)
Check. A probability of 0 means 0 out of any number of outcomes. ✓
Method. An even chance means it is as likely to happen as not to happen.
- Split the whole scale into two equal halves. Half of 1 is the middle of the scale.1 ÷ 2 = 1/2
- Write it in other forms. A half is 0.5 and 50%.1/2 = 0.5 = 50%
Check. 1/2 + 1/2 = 1, so the two halves make the whole. ✓
Method. Certain means it will definitely happen.
- Recall the end points of the scale. The scale runs from 0 to 1.0 = impossible, 1 = certain
- Choose the one for “must happen”. Probability = 1 (or 100%)
Check. All the probabilities in an experiment add up to 1. ✓
Method. Probabilities can be written as fractions, decimals or percentages. Change the fraction by making the denominator 100 or by dividing.
- Start with the fraction. A fraction, a decimal and a percentage are three ways of writing the same number.35
- Make the denominator 100. A percentage is a number out of 100. Multiply the top and the bottom by the same number.5 × 20 = 100, so 3 × 20 = 60
35 = 60100 - Write it as a decimal and a percentage. 60 out of 100 is 60%, and 60 hundredths is 0.6.60100 = 0.6 = 60%
Check. Multiply the decimal by the denominator: 0.6 × 5 = 3, which is the numerator 3. ✓
Watch out. A probability as a percentage can be written 25%, as a decimal 0.25 and as a fraction 14. They are all the same number.
Method. Probabilities can be written as fractions, decimals or percentages. Change the fraction by making the denominator 100 or by dividing.
- Start with the fraction. A fraction, a decimal and a percentage are three ways of writing the same number.720
- Make the denominator 100. A percentage is a number out of 100. Multiply the top and the bottom by the same number.20 × 5 = 100, so 7 × 5 = 35
720 = 35100 - Write it as a decimal and a percentage. 35 out of 100 is 35%, and 35 hundredths is 0.35.35100 = 0.35 = 35%
Check. Multiply the decimal by the denominator: 0.35 × 20 = 7, which is the numerator 7. ✓
Watch out. A probability as a percentage can be written 25%, as a decimal 0.25 and as a fraction 14. They are all the same number.
Method. Probabilities can be written as fractions, decimals or percentages. Change the fraction by making the denominator 100 or by dividing.
- Start with the fraction. A fraction, a decimal and a percentage are three ways of writing the same number.18
- Divide the top by the bottom to get the decimal. A fraction bar means divide.1 ÷ 8 = 0.125
- Multiply by 100 to get the percentage. A percentage is a number out of 100, so multiply the decimal by 100.0.125 × 100 = 12.5%
Check. Multiply the decimal by the denominator: 0.125 × 8 = 1, which is the numerator 1. ✓
Watch out. A probability as a percentage can be written 25%, as a decimal 0.25 and as a fraction 14. They are all the same number.
Method. Probabilities can be written as fractions, decimals or percentages. Change the fraction by making the denominator 100 or by dividing.
- Start with the fraction. A fraction, a decimal and a percentage are three ways of writing the same number.925
- Make the denominator 100. A percentage is a number out of 100. Multiply the top and the bottom by the same number.25 × 4 = 100, so 9 × 4 = 36
925 = 36100 - Write it as a decimal and a percentage. 36 out of 100 is 36%, and 36 hundredths is 0.36.36100 = 0.36 = 36%
Check. Multiply the decimal by the denominator: 0.36 × 25 = 9, which is the numerator 9. ✓
Watch out. A probability as a percentage can be written 25%, as a decimal 0.25 and as a fraction 14. They are all the same number.
Method. Probabilities can be written as fractions, decimals or percentages. Change the fraction by making the denominator 100 or by dividing.
- Start with the fraction. A fraction, a decimal and a percentage are three ways of writing the same number.34
- Make the denominator 100. A percentage is a number out of 100. Multiply the top and the bottom by the same number.4 × 25 = 100, so 3 × 25 = 75
34 = 75100 - Write it as a decimal and a percentage. 75 out of 100 is 75%, and 75 hundredths is 0.75.75100 = 0.75 = 75%
Check. Multiply the decimal by the denominator: 0.75 × 4 = 3, which is the numerator 3. ✓
Watch out. A probability as a percentage can be written 25%, as a decimal 0.25 and as a fraction 14. They are all the same number.
Method. Probabilities can be written as fractions, decimals or percentages. Change the fraction by making the denominator 100 or by dividing.
- Start with the fraction. A fraction, a decimal and a percentage are three ways of writing the same number.1150
- Make the denominator 100. A percentage is a number out of 100. Multiply the top and the bottom by the same number.50 × 2 = 100, so 11 × 2 = 22
1150 = 22100 - Write it as a decimal and a percentage. 22 out of 100 is 22%, and 22 hundredths is 0.22.22100 = 0.22 = 22%
Check. Multiply the decimal by the denominator: 0.22 × 50 = 11, which is the numerator 11. ✓
Watch out. A probability as a percentage can be written 25%, as a decimal 0.25 and as a fraction 14. They are all the same number.
Method. Work out each probability, then find the letter at that point on the scale.
- Work out P(tails). 1 of the 2 equally likely sides is tails.P(tails) = 1/2
- Find 1/2 on the scale. It is the middle of the scale.Letter B
Check. B is exactly halfway between 0 and 1. ✓
Method. Count the red sections out of all the sections.
- Work out P(red). 3 of the 4 equal sections are red.P(red) = 3/4
- Find 3/4 on the scale. It is three quarters of the way along.Letter C
Check. 3/4 = 0.75, and C is at 0.75 on the scale. ✓
Method. Count the spades out of the 52 cards.
- Work out P(spade). There are 13 spades in 52 cards.P(spade) = 13/52 = 1/4
- Find 1/4 on the scale. It is one quarter of the way along.Letter A
Check. 13 × 4 = 52, so 13/52 = 1/4. ✓
Method. A certain event has probability 1.
- Decide how likely it is. It will definitely happen, so it is certain.P = 1
- Find 1 on the scale. It is the right-hand end of the scale.Letter D
Check. Certain events have probability 1. ✓
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6
Total number of outcomes = 6 - Pick out the outcomes where the score is 5. These are the favourable outcomes. Their number goes on the top.5
Number of favourable outcomes = 1 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the score is 5) = 16
Check. The other 5 outcomes do not fit the event, so P(the event does not happen) = 56. Then 16 + 56 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6
Total number of outcomes = 6 - Pick out the outcomes where the score is even. These are the favourable outcomes. Their number goes on the top.2, 4, 6
Number of favourable outcomes = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the score is even) = 36
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 3. The value does not change but the numbers are smaller.3 ÷ 3 = 1 and 6 ÷ 3 = 2
36 = 12
Check. The other 3 outcomes do not fit the event, so P(the event does not happen) = 12. Then 36 + 36 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6
Total number of outcomes = 6 - Pick out the outcomes where the score is bigger than 4. These are the favourable outcomes. Their number goes on the top.5, 6
Number of favourable outcomes = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the score is bigger than 4) = 26
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 2. The value does not change but the numbers are smaller.2 ÷ 2 = 1 and 6 ÷ 2 = 3
26 = 13
Check. The other 4 outcomes do not fit the event, so P(the event does not happen) = 23. Then 26 + 46 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6
Total number of outcomes = 6 - Pick out the outcomes where the score is a multiple of 3. These are the favourable outcomes. Their number goes on the top.3, 6
Number of favourable outcomes = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the score is a multiple of 3) = 26
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 2. The value does not change but the numbers are smaller.2 ÷ 2 = 1 and 6 ÷ 2 = 3
26 = 13
Check. The other 4 outcomes do not fit the event, so P(the event does not happen) = 23. Then 26 + 46 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6
Total number of outcomes = 6 - Pick out the outcomes where the score is less than 7. These are the favourable outcomes. Their number goes on the top.1, 2, 3, 4, 5, 6
Number of favourable outcomes = 6 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the score is less than 7) = 66
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 6. The value does not change but the numbers are smaller.6 ÷ 6 = 1 and 6 ÷ 6 = 1
66 = 1 - Say what the answer means. Every outcome fits the event, so it is certain to happen. A certain event has probability 1.P(the score is less than 7) = 1
Check. The other 0 outcomes do not fit the event, so P(the event does not happen) = 0. Then 66 + 06 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6
Total number of outcomes = 6 - Pick out the outcomes where the score is 7. These are the favourable outcomes. Their number goes on the top.none
Number of favourable outcomes = 0 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the score is 7) = 06
- Say what the answer means. No outcome fits the event, so it cannot happen. An impossible event has probability 0.P(the score is 7) = 0
Check. The other 6 outcomes do not fit the event, so P(the event does not happen) = 1. Then 06 + 66 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of counters. Every counter is equally likely to be picked, so add up all the counters. This is the bottom of the fraction.4 + 3 + 5 = 12
- Count the counters that are red. These are the favourable outcomes. This number goes on the top.4 red counters = 4
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the counter is red) = 412
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.4 ÷ 4 = 1 and 12 ÷ 4 = 3
412 = 13
Check. The other 8 outcomes do not fit the event, so P(the event does not happen) = 23. Then 412 + 812 = 1212 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of counters. Every counter is equally likely to be picked, so add up all the counters. This is the bottom of the fraction.4 + 3 + 5 = 12
- Count the counters that are blue. These are the favourable outcomes. This number goes on the top.3 blue counters = 3
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the counter is blue) = 312
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 3. The value does not change but the numbers are smaller.3 ÷ 3 = 1 and 12 ÷ 3 = 4
312 = 14
Check. The other 9 outcomes do not fit the event, so P(the event does not happen) = 34. Then 312 + 912 = 1212 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of counters. Every counter is equally likely to be picked, so add up all the counters. This is the bottom of the fraction.4 + 3 + 5 = 12
- Count the counters that are green. These are the favourable outcomes. This number goes on the top.5 green counters = 5
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the counter is green) = 512
Check. The other 7 outcomes do not fit the event, so P(the event does not happen) = 712. Then 512 + 712 = 1212 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total number of counters. Every counter is equally likely to be picked, so add up all the counters. This is the bottom of the fraction.4 + 3 + 5 = 12
- Count the counters that are red and blue. These are the favourable outcomes. This number goes on the top.4 + 3 = 7
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the counter is red or blue) = 712
Check. The other 5 outcomes do not fit the event, so P(the event does not happen) = 512. Then 712 + 512 = 1212 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.red, blue, green, red, yellow, red
Total number of sections = 6 - Pick out the outcomes where the spinner lands on red. These are the favourable outcomes. Their number goes on the top.red, red, red
Number of favourable sections = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the spinner lands on red) = 36
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 3. The value does not change but the numbers are smaller.3 ÷ 3 = 1 and 6 ÷ 3 = 2
36 = 12
Check. The other 3 outcomes do not fit the event, so P(the event does not happen) = 12. Then 36 + 36 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.red, blue, green, red, yellow, red
Total number of sections = 6 - Pick out the outcomes where the spinner lands on blue. These are the favourable outcomes. Their number goes on the top.blue
Number of favourable sections = 1 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the spinner lands on blue) = 16
Check. The other 5 outcomes do not fit the event, so P(the event does not happen) = 56. Then 16 + 56 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.red, blue, green, red, yellow, red
Total number of sections = 6 - Pick out the outcomes where the spinner lands on green or yellow. These are the favourable outcomes. Their number goes on the top.green, yellow
Number of favourable sections = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the spinner lands on green or yellow) = 26
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 2. The value does not change but the numbers are smaller.2 ÷ 2 = 1 and 6 ÷ 2 = 3
26 = 13
Check. The other 4 outcomes do not fit the event, so P(the event does not happen) = 23. Then 26 + 46 = 66 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Method. Put the red counters over all the counters, then change to a decimal.
- Find the total. The bottom of the fraction is all the counters.3 + 7 = 10
- Write P(red). Red counters over all counters.P(red) = 3/10
- Write it as a decimal. Divide 3 by 10.3 ÷ 10 = 0.3
Check. 0.3 = 30%, and 30% of 10 counters is 3 counters. ✓
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.M, A, T, H, E, M, A, T, I, C, S
Total number of letters = 11 - Pick out the outcomes where the letter is M. These are the favourable outcomes. Their number goes on the top.M, M
Number of favourable letters = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the letter is M) = 211
Check. The other 9 outcomes do not fit the event, so P(the event does not happen) = 911. Then 211 + 911 = 1111 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.M, A, T, H, E, M, A, T, I, C, S
Total number of letters = 11 - Pick out the outcomes where the letter is a vowel. These are the favourable outcomes. Their number goes on the top.A, E, A, I
Number of favourable letters = 4 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the letter is a vowel) = 411
Check. The other 7 outcomes do not fit the event, so P(the event does not happen) = 711. Then 411 + 711 = 1111 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.M, A, T, H, E, M, A, T, I, C, S
Total number of letters = 11 - Pick out the outcomes where the letter is T or C. These are the favourable outcomes. Their number goes on the top.T, T, C
Number of favourable letters = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the letter is T or C) = 311
Check. The other 8 outcomes do not fit the event, so P(the event does not happen) = 811. Then 311 + 811 = 1111 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.M, A, T, H, E, M, A, T, I, C, S
Total number of letters = 11 - Pick out the outcomes where the letter is Z. These are the favourable outcomes. Their number goes on the top.none
Number of favourable letters = 0 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the letter is Z) = 011
- Say what the answer means. No outcome fits the event, so it cannot happen. An impossible event has probability 0.P(the letter is Z) = 0
Check. The other 11 outcomes do not fit the event, so P(the event does not happen) = 1. Then 011 + 1111 = 1111 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. A standard pack has 4 suits (hearts, diamonds, clubs, spades) with 13 cards in each suit, so every card is equally likely.4 × 13 = 52 cards
- Find the number we want. Count the cards in the pack that match.13 hearts, so 13
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the card is a heart) = 1352
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 13. The value does not change but the numbers are smaller.13 ÷ 13 = 1 and 52 ÷ 13 = 4
1352 = 14
Check. The other 39 outcomes do not fit the event, so P(the event does not happen) = 34. Then 1352 + 3952 = 5252 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. A standard pack has 4 suits (hearts, diamonds, clubs, spades) with 13 cards in each suit, so every card is equally likely.4 × 13 = 52 cards
- Find the number we want. Count the cards in the pack that match.one king in each suit: 4 × 1 = 4
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the card is a king) = 452
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.4 ÷ 4 = 1 and 52 ÷ 4 = 13
452 = 113
Check. The other 48 outcomes do not fit the event, so P(the event does not happen) = 1213. Then 452 + 4852 = 5252 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. A standard pack has 4 suits (hearts, diamonds, clubs, spades) with 13 cards in each suit, so every card is equally likely.4 × 13 = 52 cards
- Find the number we want. Count the cards in the pack that match.the queen of hearts and the queen of diamonds: 2
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the card is a red queen) = 252
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 2. The value does not change but the numbers are smaller.2 ÷ 2 = 1 and 52 ÷ 2 = 26
252 = 126
Check. The other 50 outcomes do not fit the event, so P(the event does not happen) = 2526. Then 252 + 5052 = 5252 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. A standard pack has 4 suits (hearts, diamonds, clubs, spades) with 13 cards in each suit, so every card is equally likely.4 × 13 = 52 cards
- Find the number we want. Count the cards in the pack that match.3 picture cards in each of 4 suits: 3 × 4 = 12
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the card is a picture card) = 1252
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.12 ÷ 4 = 3 and 52 ÷ 4 = 13
1252 = 313
Check. The other 40 outcomes do not fit the event, so P(the event does not happen) = 1013. Then 1252 + 4052 = 5252 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. A standard pack has 4 suits (hearts, diamonds, clubs, spades) with 13 cards in each suit, so every card is equally likely.4 × 13 = 52 cards
- Find the number we want. Count the cards in the pack that match.only 1 card matches
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the card is the ace of spades) = 152
Check. The other 51 outcomes do not fit the event, so P(the event does not happen) = 5152. Then 152 + 5152 = 5252 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Total number of numbers = 20 - Pick out the outcomes where the number is a multiple of 5. These are the favourable outcomes. Their number goes on the top.5, 10, 15, 20
Number of favourable numbers = 4 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the number is a multiple of 5) = 420
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.4 ÷ 4 = 1 and 20 ÷ 4 = 5
420 = 15
Check. The other 16 outcomes do not fit the event, so P(the event does not happen) = 45. Then 420 + 1620 = 2020 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Total number of numbers = 20 - Pick out the outcomes where the number is a square number. These are the favourable outcomes. Their number goes on the top.1, 4, 9, 16
Number of favourable numbers = 4 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the number is a square number) = 420
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.4 ÷ 4 = 1 and 20 ÷ 4 = 5
420 = 15
Check. The other 16 outcomes do not fit the event, so P(the event does not happen) = 45. Then 420 + 1620 = 2020 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Total number of numbers = 20 - Pick out the outcomes where the number is a factor of 20. These are the favourable outcomes. Their number goes on the top.1, 2, 4, 5, 10, 20
Number of favourable numbers = 6 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the number is a factor of 20) = 620
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 2. The value does not change but the numbers are smaller.6 ÷ 2 = 3 and 20 ÷ 2 = 10
620 = 310
Check. The other 14 outcomes do not fit the event, so P(the event does not happen) = 710. Then 620 + 1420 = 2020 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Total number of numbers = 20 - Pick out the outcomes where the number is prime. These are the favourable outcomes. Their number goes on the top.2, 3, 5, 7, 11, 13, 17, 19
Number of favourable numbers = 8 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the number is prime) = 820
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.8 ÷ 4 = 2 and 20 ÷ 4 = 5
820 = 25
Check. The other 12 outcomes do not fit the event, so P(the event does not happen) = 35. Then 820 + 1220 = 2020 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. Every ticket has the same chance of being drawn, so the bottom is all the tickets sold.250 tickets
- Find the number we want. Sam has 10 of them.10 tickets
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(Sam wins) = 10250
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 10. The value does not change but the numbers are smaller.10 ÷ 10 = 1 and 250 ÷ 10 = 25
10250 = 125
Check. The other 240 outcomes do not fit the event, so P(the event does not happen) = 2425. Then 10250 + 240250 = 250250 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. Every ticket has the same chance of being drawn, so the bottom is all the tickets sold.250 tickets
- Find the number we want. Tia has 15 of them.15 tickets
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(Tia wins) = 15250
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 5. The value does not change but the numbers are smaller.15 ÷ 5 = 3 and 250 ÷ 5 = 50
15250 = 350
Check. The other 235 outcomes do not fit the event, so P(the event does not happen) = 4750. Then 15250 + 235250 = 250250 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. Every ticket has the same chance of being drawn, so the bottom is all the tickets sold.250 tickets
- Find the number we want. Sam and Tia hold 10 + 15 tickets between them.10 + 15 = 25 tickets
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(Sam or Tia wins) = 25250
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 25. The value does not change but the numbers are smaller.25 ÷ 25 = 1 and 250 ÷ 25 = 10
25250 = 110
Check. The other 225 outcomes do not fit the event, so P(the event does not happen) = 910. Then 25250 + 225250 = 250250 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Method. Work out each probability, write them with a common denominator, then put them in order.
- Work out each probability. Use favourable ÷ total each time.A: 1/2
B: 1/6
C: 4/6 = 2/3
D: 3/12 = 1/4 - Write them with the same denominator. The lowest common denominator of 2, 6, 3 and 4 is 12, so we can compare the numerators.A: 6/12
B: 2/12
C: 8/12
D: 3/12 - Put the numerators in order, smallest first. The smaller the probability, the less likely the event.2, 3, 6, 8 so B, D, A, C
Check. As decimals: B = 0.167, D = 0.25, A = 0.5, C = 0.667, which are in increasing order. ✓
Watch out. Compare fractions only after giving them the same denominator.
Method. The rule favourable ÷ total only works when all the outcomes are equally likely. Check whether that is true here.
- Find the two outcomes Ben has used. Ben says there are two outcomes: rain, or no rain.Outcomes: rain, no rain
- Ask whether they are equally likely. The chance of rain depends on the weather and the season, so there is no reason for it to be exactly one half just because there are two outcomes.The outcomes are not equally likely.
- Write the conclusion. Having two outcomes does not make each one 1/2.Ben is wrong: P(rain) is not 1/2 just because there are two outcomes.
Check. Compare with a die: it has 6 outcomes but P(6) = 1/6 only because they are equally likely. If a die were loaded, this would fail too. ✓
Watch out. Only use favourable ÷ total when every outcome has the same chance.
Method. Think about what a probability tells us and what it does not.
- Read what P(6) = 1/6 tells us. It tells us how likely a 6 is on each roll. It does not tell us what will happen in a short run of rolls.On each roll, P(6) = 1/6
- Think about 6 rolls. Each roll is separate and random. You could roll no sixes at all, or two sixes, or more.The number of sixes in 6 rolls can be 0, 1, 2, 3, ...
- Conclude. About one 6 in every six rolls is what we expect over many rolls, but it is not guaranteed.No, Zara is not correct.
Check. If it were true, then rolling 12 times would give exactly 2 sixes every time, which does not happen in practice. ✓
Watch out. Probability describes what is likely over many trials, not what must happen in a few.
Method. Probability = favourable outcomes ÷ all the counters, not ÷ the number of colours.
- Count all the counters. Each counter is an equally likely outcome. There are three red and one blue.3 + 1 = 4 counters
- Find the favourable outcomes. Only one counter is blue.1 blue counter
- Write the correct probability. Blue counters over all counters.P(blue) = 14
- Say what Jack did wrong. He counted colours (2) instead of counters (4), but the colours are not equally likely because there are 3 red counters.Jack should use 4, not 2.
Check. P(red) = 3/4 and 3/4 + 1/4 = 1. ✓
Watch out. Count the objects, not the types of object.
Method. The three colours together make all 12 sections, so take the red and blue away from 12.
- Add the red and blue sections. These are the sections that are not green.3 + 4 = 7
- Take the total from 12. The rest are green.12 − 7 = 5
Check. 3 + 4 + 5 = 12. ✓
Watch out. Check your three numbers add up to the total number of sections.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total. All 12 sections are equally likely.12 sections
- Find the number we want. There are 5 green sections from part (a).5 sections
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the spinner lands on green) = 512
Check. The other 7 outcomes do not fit the event, so P(the event does not happen) = 712. Then 512 + 712 = 1212 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. Write the answer as a fraction, decimal or percentage, never as a ratio such as 1 : 6. A fraction is best left in its simplest form.
Method. Use the probability to find the total number of counters, then take away the red ones.
- Write what the probability means. P(red) = red counters ÷ all counters, and P(red) = 1/5, so the red counters are one fifth of all the counters.3 red counters are 15 of the total
- Find the total. If 3 counters are 1 fifth, then the whole is 5 times as many.3 × 5 = 15 counters
- Find the blue counters. Blue = total − red.15 − 3 = 12
Check. P(red) = 3/15 = 1/5, which matches the question. ✓
Watch out. A probability of 1/5 does not mean there are 5 counters. It means 1 in every 5, so scale up from the known number.
Method. Remember the range of a probability.
- Recall the scale. Probabilities run from 0 (impossible) to 1 (certain).0 ≤ probability ≤ 1
- Compare 1.2. 1.2 is bigger than 1, which would be “more than certain”, and that does not make sense.1.2 > 1
- Conclude. So 1.2 cannot be a probability.
Check. A fraction of the outcomes can never be more than all of the outcomes, so the fraction is at most 1. ✓