GCSE Maths: Listing outcomes and sample-space tables (Higher)
Part of the GCSE Maths probability chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Listing outcomes and sample-space tables
To use the rule probability = favourable ÷ total we must know all the outcomes. When there are two or more events, such as tossing two coins, it is easy to miss one. The cure is to list the outcomes systematically: in a fixed order, so that nothing is missed or repeated.
Listing. For two coins, write what the first coin shows, then every possibility for the second: first coin heads (H): HH, HT. First coin tails (T): TH, TT. That gives 4 outcomes: HH, HT, TH, TT. Notice that HT and TH are different outcomes (the first coin and the second coin are different), and each of the four is equally likely. So P(two heads) = 1/4, and P(one head and one tail) = 2/4 = 1/2, not 1/3.
| coin 2 \ coin 1 | H | T |
|---|---|---|
| H | HH | HT |
| T | TH | TT |
Sample-space tables. For two events with several outcomes each, a table (a sample space diagram) is neat. Put the outcomes of one event along the top and the other down the side. Each cell gives one combined outcome, such as the total of two dice. Count the cells that match the question, and count all the cells. For two dice there are 6 × 6 = 36 equally likely outcomes.
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Counting without listing. Sometimes we only need to know how many outcomes there are. If there are 3 starters and 2 main courses, each starter can go with each main course, so there are 3 × 2 = 6 different two-course meals. This is the product rule for counting: multiply the number of choices at each stage. It also tells us the size of the sample space: two dice give 6 × 6 = 36, three coins give 2 × 2 × 2 = 8.
Key ideas
- List outcomes in a fixed order so none is missed or repeated. HT and TH are different outcomes.
- A sample-space table shows every combined outcome of two events as a cell.
- Probability = number of matching cells ÷ total number of cells.
- Two dice have 36 outcomes. Two coins have 4. Three coins have 8.
- Product rule: number of outcomes = (choices at stage 1) × (choices at stage 2) × ...
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. Write the first coin first, then the second.HH, HT, TH, TT
Total number of outcomes = 4 - Pick out the outcomes where one head and one tail. These are the favourable outcomes. Their number goes on the top.HT, TH
Number of favourable outcomes = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(one head and one tail) = 24
- 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 4 ÷ 2 = 2
24 = 12
Check. The other 2 outcomes do not fit the event, so P(the event does not happen) = 12. Then 24 + 24 = 44 = 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.
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely. Each cell shows the total of the two scores.
Total number of outcomes = 6 × 6 = 36+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Highlight the cells where the total is 8. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 5+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is 8) = 536
Check. The other 31 outcomes do not fit the event, so P(the event does not happen) = 3136. Then 536 + 3136 = 3636 = 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. List spinner A first, then spinner B: for each number there are two colours.1R, 1B, 2R, 2B, 3R, 3B
Total number of outcomes = 6 - Pick out the outcomes where an odd number and red. These are the favourable outcomes. Their number goes on the top.1R, 3R
Number of favourable outcomes = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(an odd number and red) = 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. To count how many outcomes there are, multiply the number of choices at each stage.
- Split the problem into stages. We make one choice at a time, so we count the choices stage by stage.Sandwiches then Drinks
- Write down the number of choices at each stage. For each stage we count how many choices there are.Sandwiches: 3 choices
Drinks: 4 choices - Multiply the numbers of choices. Each choice at one stage can be paired with every choice at the next stage, so the numbers multiply (this is the product rule for counting).3 × 4 = 12
Check. For two stages, picture a table with 3 rows and 4 columns: it has 3 × 4 = 12 cells. ✓
Watch out. Multiply the numbers of choices. Do not add them.
Watch out
Do not forget that HT and TH are different outcomes. If you count “one head and one tail” as a single outcome you get 3 outcomes and the wrong answer 1/3. Also be careful to count all the cells, not just the highlighted ones, for the bottom of the fraction.Exercise 23E Listing outcomes and sample-space tables
Non-calculatorMethod. To count how many outcomes there are, multiply the number of choices at each stage.
- Split the problem into stages. We make one choice at a time, so we count the choices stage by stage.Coin then Die
- Write down the number of choices at each stage. For each stage we count how many choices there are.Coin: 2 choices
Die: 6 choices - Multiply the numbers of choices. Each choice at one stage can be paired with every choice at the next stage, so the numbers multiply (this is the product rule for counting).2 × 6 = 12
Check. For two stages, picture a table with 2 rows and 6 columns: it has 2 × 6 = 12 cells. ✓
Watch out. Multiply the numbers of choices. Do not add them.
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.H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6
Total number of outcomes = 12 - Pick out the outcomes where heads and a 6. These are the favourable outcomes. Their number goes on the top.H6
Number of favourable outcomes = 1 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(heads and a 6) = 112
Check. The other 11 outcomes do not fit the event, so P(the event does not happen) = 1112. Then 112 + 1112 = 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.H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6
Total number of outcomes = 12 - Pick out the outcomes where tails and an odd number. These are the favourable outcomes. Their number goes on the top.T1, T3, T5
Number of favourable outcomes = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(tails and an odd number) = 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.
- List all the possible outcomes. The total number of equally likely outcomes goes on the bottom of the probability fraction.HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Total number of outcomes = 8 - Pick out the outcomes where exactly two heads. These are the favourable outcomes. Their number goes on the top.HHT, HTH, THH
Number of favourable outcomes = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(exactly two heads) = 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.
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.HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Total number of outcomes = 8 - Pick out the outcomes where at least one tail. These are the favourable outcomes. Their number goes on the top.HHT, HTH, HTT, THH, THT, TTH, TTT
Number of favourable outcomes = 7 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(at least one tail) = 78
Check. The other 1 outcome do not fit the event, so P(the event does not happen) = 18. Then 78 + 18 = 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.
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.HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Total number of outcomes = 8 - Pick out the outcomes where all three coins the same. These are the favourable outcomes. Their number goes on the top.HHH, TTT
Number of favourable outcomes = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(all three coins the same) = 28
- 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 8 ÷ 2 = 4
28 = 14
Check. The other 6 outcomes do not fit the event, so P(the event does not happen) = 34. Then 28 + 68 = 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.
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely.
Total number of outcomes = 6 × 6 = 36+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Highlight the cells where the total is 7. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 6+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is 7) = 636
- 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 36 ÷ 6 = 6
636 = 16
Check. The other 30 outcomes do not fit the event, so P(the event does not happen) = 56. Then 636 + 3036 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely.
Total number of outcomes = 6 × 6 = 36+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Highlight the cells where the total is 12. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 1+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is 12) = 136
Check. The other 35 outcomes do not fit the event, so P(the event does not happen) = 3536. Then 136 + 3536 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely.
Total number of outcomes = 6 × 6 = 36+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Highlight the cells where the total is less than 5. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 6+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is less than 5) = 636
- 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 36 ÷ 6 = 6
636 = 16
Check. The other 30 outcomes do not fit the event, so P(the event does not happen) = 56. Then 636 + 3036 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely.
Total number of outcomes = 6 × 6 = 36+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Highlight the cells where the total is even. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 18+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is even) = 1836
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 18. The value does not change but the numbers are smaller.18 ÷ 18 = 1 and 36 ÷ 18 = 2
1836 = 12
Check. The other 18 outcomes do not fit the event, so P(the event does not happen) = 12. Then 1836 + 1836 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely.
Total number of outcomes = 6 × 6 = 36+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Highlight the cells where the total is a prime number. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 15+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is a prime number) = 1536
- 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.15 ÷ 3 = 5 and 36 ÷ 3 = 12
1536 = 512
Check. The other 21 outcomes do not fit the event, so P(the event does not happen) = 712. Then 1536 + 2136 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely. Each cell shows the difference (bigger − smaller).
Total number of outcomes = 6 × 6 = 36− 1 2 3 4 5 6 1 0 1 2 3 4 5 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 5 4 3 2 1 0 1 6 5 4 3 2 1 0 - Highlight the cells where the difference is 0. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 6− 1 2 3 4 5 6 1 0 1 2 3 4 5 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 5 4 3 2 1 0 1 6 5 4 3 2 1 0 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the difference is 0) = 636
- 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 36 ÷ 6 = 6
636 = 16
Check. The other 30 outcomes do not fit the event, so P(the event does not happen) = 56. Then 636 + 3036 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely. Each cell shows the difference (bigger − smaller).
Total number of outcomes = 6 × 6 = 36− 1 2 3 4 5 6 1 0 1 2 3 4 5 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 5 4 3 2 1 0 1 6 5 4 3 2 1 0 - Highlight the cells where the difference is at least 4. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 6− 1 2 3 4 5 6 1 0 1 2 3 4 5 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 5 4 3 2 1 0 1 6 5 4 3 2 1 0 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the difference is at least 4) = 636
- 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 36 ÷ 6 = 6
636 = 16
Check. The other 30 outcomes do not fit the event, so P(the event does not happen) = 56. Then 636 + 3036 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely. Each cell shows the difference (bigger − smaller).
Total number of outcomes = 6 × 6 = 36− 1 2 3 4 5 6 1 0 1 2 3 4 5 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 5 4 3 2 1 0 1 6 5 4 3 2 1 0 - Highlight the cells where the difference is 1. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 10− 1 2 3 4 5 6 1 0 1 2 3 4 5 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 5 4 3 2 1 0 1 6 5 4 3 2 1 0 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the difference is 1) = 1036
- 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.10 ÷ 2 = 5 and 36 ÷ 2 = 18
1036 = 518
Check. The other 26 outcomes do not fit the event, so P(the event does not happen) = 1318. Then 1036 + 2636 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely. Each cell shows the product.
Total number of outcomes = 6 × 6 = 36× 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36 - Highlight the cells where the product is 12. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 4× 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the product is 12) = 436
- 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 36 ÷ 4 = 9
436 = 19
Check. The other 32 outcomes do not fit the event, so P(the event does not happen) = 89. Then 436 + 3236 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely. Each cell shows the product.
Total number of outcomes = 6 × 6 = 36× 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36 - Highlight the cells where the product is odd. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 9× 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the product is odd) = 936
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 9. The value does not change but the numbers are smaller.9 ÷ 9 = 1 and 36 ÷ 9 = 4
936 = 14
Check. The other 27 outcomes do not fit the event, so P(the event does not happen) = 34. Then 936 + 2736 = 3636 = 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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely. Each cell shows the product.
Total number of outcomes = 6 × 6 = 36× 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36 - Highlight the cells where the product is more than 20. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 6× 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the product is more than 20) = 636
- 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 36 ÷ 6 = 6
636 = 16
Check. The other 30 outcomes do not fit the event, so P(the event does not happen) = 56. Then 636 + 3036 = 3636 = 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.
- Draw the sample space table. Write the spinner A scores along the top and the spinner B scores down the side. Every cell is one outcome, and all the cells are equally likely.
Total number of outcomes = 4 × 3 = 12+ 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 - Highlight the cells where the total is 5. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 3+ 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is 5) = 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.
- Draw the sample space table. Write the spinner A scores along the top and the spinner B scores down the side. Every cell is one outcome, and all the cells are equally likely.
Total number of outcomes = 4 × 3 = 12+ 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 - Highlight the cells where the total is more than 4. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 6+ 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is more than 4) = 612
- 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 12 ÷ 6 = 2
612 = 12
Check. The other 6 outcomes do not fit the event, so P(the event does not happen) = 12. Then 612 + 612 = 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.
- Draw the sample space table. Write the spinner A scores along the top and the spinner B scores down the side. Every cell is one outcome, and all the cells are equally likely.
Total number of outcomes = 4 × 3 = 12+ 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 - Highlight the cells where the total is even. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 6+ 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is even) = 612
- 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 12 ÷ 6 = 2
612 = 12
Check. The other 6 outcomes do not fit the event, so P(the event does not happen) = 12. Then 612 + 612 = 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.11, 12, 13, 21, 22, 23, 31, 32, 33
Total number of numbers = 9 - Pick out the outcomes where the number is even. These are the favourable outcomes. Their number goes on the top.12, 22, 32
Number of favourable numbers = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the number is even) = 39
- 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 9 ÷ 3 = 3
39 = 13
Check. The other 6 outcomes do not fit the event, so P(the event does not happen) = 23. Then 39 + 69 = 99 = 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.11, 12, 13, 21, 22, 23, 31, 32, 33
Total number of numbers = 9 - Pick out the outcomes where the number is bigger than 25. These are the favourable outcomes. Their number goes on the top.31, 32, 33
Number of favourable numbers = 3 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the number is bigger than 25) = 39
- 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 9 ÷ 3 = 3
39 = 13
Check. The other 6 outcomes do not fit the event, so P(the event does not happen) = 23. Then 39 + 69 = 99 = 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. List systematically: all the orders starting with A, then B, then C.ABC, ACB, BAC, BCA, CAB, CBA
Total number of outcomes = 6 - Pick out the outcomes where A is first. These are the favourable outcomes. Their number goes on the top.ABC, ACB
Number of favourable outcomes = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(A is first) = 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. To count how many outcomes there are, multiply the number of choices at each stage.
- Split the problem into stages. We make one choice at a time, so we count the choices stage by stage.Sandwiches then Soups
- Write down the number of choices at each stage. For each stage we count how many choices there are.Sandwiches: 3 choices
Soups: 2 choices - Multiply the numbers of choices. Each choice at one stage can be paired with every choice at the next stage, so the numbers multiply (this is the product rule for counting).3 × 2 = 6
Check. For two stages, picture a table with 3 rows and 2 columns: it has 3 × 2 = 6 cells. ✓
Watch out. Multiply the numbers of choices. Do not add them.
Method. To count how many outcomes there are, multiply the number of choices at each stage.
- Split the problem into stages. We make one choice at a time, so we count the choices stage by stage.Sandwiches then Soups then Drinks
- Write down the number of choices at each stage. For each stage we count how many choices there are.Sandwiches: 3 choices
Soups: 2 choices
Drinks: 4 choices - Multiply the numbers of choices. Each choice at one stage can be paired with every choice at the next stage, so the numbers multiply (this is the product rule for counting).3 × 2 × 4 = 24
Check. Multiply in the opposite order: 4 × 2 × 3 = 24. ✓
Watch out. Multiply the numbers of choices. Do not add them.
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.cheese, tomato, cheese, leek, ham, tomato, ham, leek, egg, tomato, egg, leek
Total number of outcomes = 6 - Pick out the outcomes where ham and tomato. These are the favourable outcomes. Their number goes on the top.ham, tomato
Number of favourable outcomes = 1 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(ham and tomato) = 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.A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3, C4
Total number of codes = 12 - Pick out the outcomes where A followed by an even digit. These are the favourable outcomes. Their number goes on the top.A2, A4
Number of favourable codes = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(A followed by an even digit) = 212
- 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 12 ÷ 2 = 6
212 = 16
Check. The other 10 outcomes do not fit the event, so P(the event does not happen) = 56. Then 212 + 1012 = 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.A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3, C4
Total number of codes = 12 - Pick out the outcomes where B3 or C3. These are the favourable outcomes. Their number goes on the top.B3, C3
Number of favourable codes = 2 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(B3 or C3) = 212
- 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 12 ÷ 2 = 6
212 = 16
Check. The other 10 outcomes do not fit the event, so P(the event does not happen) = 56. Then 212 + 1012 = 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. A game is fair if both players have the same probability of winning. Find each probability from the sample-space table.
- Count the outcomes. Two dice have 6 × 6 outcomes in the table.6 × 6 = 36
- Count the totals of 8 or more. Totals: 8 occurs 5 times, 9 occurs 4 times, 10 occurs 3 times, 11 occurs 2 times, 12 occurs 1 time.5 + 4 + 3 + 2 + 1 = 15
- Find the probabilities. Ali wins on 15 cells. Beth wins on the rest.P(Ali) = 1536 = 512
P(Beth) = 2136 = 712 - Compare. The probabilities are not equal.Beth is more likely to win, so the game is not fair.
Check. 5/12 + 7/12 = 1, and a total of 7 or less has 6 + 5 + 4 + 3 + 2 + 1 = 21 cells. ✓
Watch out. Fair means equal probabilities, so compare the probabilities, not who “feels” more likely.
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.
- Draw the sample space table. Write the red die scores along the top and the blue die scores down the side. Every cell is one outcome, and all the cells are equally likely. Write each cell as (red, blue).
Total number of outcomes = 6 × 6 = 36red \ blue 1 2 3 4 5 6 1 1,1 2,1 3,1 4,1 5,1 6,1 2 1,2 2,2 3,2 4,2 5,2 6,2 3 1,3 2,3 3,3 4,3 5,3 6,3 4 1,4 2,4 3,4 4,4 5,4 6,4 5 1,5 2,5 3,5 4,5 5,5 6,5 6 1,6 2,6 3,6 4,6 5,6 6,6 - Highlight the cells where red is bigger than blue. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 15red \ blue 1 2 3 4 5 6 1 1,1 2,1 3,1 4,1 5,1 6,1 2 1,2 2,2 3,2 4,2 5,2 6,2 3 1,3 2,3 3,3 4,3 5,3 6,3 4 1,4 2,4 3,4 4,4 5,4 6,4 5 1,5 2,5 3,5 4,5 5,5 6,5 6 1,6 2,6 3,6 4,6 5,6 6,6 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(red is bigger than blue) = 1536
- 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.15 ÷ 3 = 5 and 36 ÷ 3 = 12
1536 = 512
Check. The other 21 outcomes do not fit the event, so P(the event does not happen) = 712. Then 1536 + 2136 = 3636 = 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. The rule favourable ÷ total only works if the outcomes are equally likely. Check whether Tom’s three outcomes are equally likely.
- List all the outcomes properly. Treat the coins as different: first coin then second coin.HH, HT, TH, TT
- Count the outcomes for each of Tom’s results. Two heads: HH (1 way). One head: HT, TH (2 ways). No heads: TT (1 way).Not equally likely: “one head” is twice as likely
- Find the correct probability. Use the 4 equally likely outcomes.P(two heads) = 14
Check. 1/4 + 2/4 + 1/4 = 1. ✓
Watch out. Always list outcomes so that each one is equally likely.
Method. Count how many cells of the sample-space table give each total, then compare.
- Draw the sample-space table of totals. Each cell is one of 36 equally likely outcomes.
+ 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 - Count the cells for each total. Look along each diagonal of equal totals.2: 1, 3: 2, 4: 3, 5: 4, 6: 5, 7: 6, 8: 5, 9: 4, 10: 3, 11: 2, 12: 1
- Compare. The largest count is for 7.7 occurs 6 times, and every other total occurs 5 times or fewer.
- Write the probability. 6 out of 36.P(7) = 636 = 16
Check. The counts add up: 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36. ✓
Watch out. A table makes it easy to see why 7 is the most common total.
Method. To count how many outcomes there are, multiply the number of choices at each stage.
- Split the problem into stages. We make one choice at a time, so we count the choices stage by stage.First die then Second die
- Write down the number of choices at each stage. For each stage we count how many choices there are.First die: 6 choices
Second die: 6 choices - Multiply the numbers of choices. Each choice at one stage can be paired with every choice at the next stage, so the numbers multiply (this is the product rule for counting).6 × 6 = 36
Check. For two stages, picture a table with 6 rows and 6 columns: it has 6 × 6 = 36 cells. ✓
Watch out. Multiply the numbers of choices. Do not add them.
Method. To count how many outcomes there are, multiply the number of choices at each stage.
- Split the problem into stages. We make one choice at a time, so we count the choices stage by stage.First die then Second die then Third die
- Write down the number of choices at each stage. For each stage we count how many choices there are.First die: 6 choices
Second die: 6 choices
Third die: 6 choices - Multiply the numbers of choices. Each choice at one stage can be paired with every choice at the next stage, so the numbers multiply (this is the product rule for counting).6 × 6 × 6 = 216
Check. Multiply in the opposite order: 6 × 6 × 6 = 216. ✓
Watch out. Multiply the numbers of choices. Do not add them.
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.
- Draw the sample space table. Write the first die scores along the top and the second die scores down the side. Every cell is one outcome, and all the cells are equally likely. Totals that are multiples of 3 are 3 and 6.
Total number of outcomes = 4 × 4 = 16+ 1 2 3 4 1 2 3 4 5 2 3 4 5 6 3 4 5 6 7 4 5 6 7 8 - Highlight the cells where the total is a multiple of 3. These are the favourable outcomes. Their number goes on the top of the fraction.
Number of highlighted cells = 5+ 1 2 3 4 1 2 3 4 5 2 3 4 5 6 3 4 5 6 7 4 5 6 7 8 - Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(the total is a multiple of 3) = 516
Check. The other 11 outcomes do not fit the event, so P(the event does not happen) = 1116. Then 516 + 1116 = 1616 = 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.