GCSE Maths: Probabilities add up to 1: the probability of “not A” (Foundation)
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Probabilities add up to 1: the probability of “not A”
Look at a single roll of a die. Exactly one of the numbers 1, 2, 3, 4, 5 or 6 must come up. These six outcomes cover every possibility, so we call them exhaustive. They cannot happen at the same time, so we also call them mutually exclusive (the die cannot show 2 and 5 together). Because exactly one of them must happen, the probabilities of all the outcomes add up to 1: 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1. This is a very useful fact, because if we know all the probabilities but one, we can work out the missing one by taking the rest away from 1.
The complement. For any event A, the event “A does not happen” is called not A (or the complement of A, written A′). Since A either happens or does not happen, and these two cannot both happen, P(not A) = 1 − P(A). For example, if the probability that it rains is 0.3, then the probability that it does not rain is 1 − 0.3 = 0.7. It is usually quicker to work out “not A” this way than to count all the outcomes that are not A. Use it whenever you see the words not, fails, misses or at least one (you will see the last one in section 23.5).
Fractions, decimals and percentages. The rule works in every form. With a fraction, write 1 as a fraction with the same denominator: 1 − 3/8 = 8/8 − 3/8 = 5/8. With a decimal, 1 − 0.35 = 0.65. With a percentage, 100% − 35% = 65%.
Missing probabilities. If several outcomes are listed in a table and one probability is missing, add the known ones and take the total from 1. Adding for “or”. If two events cannot happen together (mutually exclusive), the probability of one or the other is found by adding their probabilities. For example, if P(red) = 0.2 and P(blue) = 0.35 for a spinner, then P(red or blue) = 0.2 + 0.35 = 0.55. We use this idea a lot in sections 23.5 and 23.6.
Key ideas
- The probabilities of all the possible outcomes of an experiment add up to 1.
- P(not A) = 1 − P(A), in any form: fractions, decimals or percentages.
- Mutually exclusive events cannot happen together. Exhaustive outcomes cover every possibility.
- To find a missing probability: add the known ones and subtract the total from 1.
- For mutually exclusive events, P(A or B) = P(A) + P(B).
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “late” happens or “not late” happens, never both, so these two events are opposites that cover every possibility.P(late) = 0.35
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not late) = 1 − P(late)
- Subtract from 1. Line up the decimal points and take the probability away from 1.1 − 0.35 = 0.65
Check. Add the two probabilities: 0.35 + 0.65 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “win” happens or “not win” happens, never both, so these two events are opposites that cover every possibility.P(win) = 38
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not win) = 1 − P(win)
- Write 1 as a fraction with the same denominator. We can only subtract fractions with the same denominator. 1 whole is 8 out of 8.1 = 88
- Subtract the numerators. The denominator stays the same.88 − 38 = 58
Check. Add the two probabilities: 38 + 58 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
| Colour | Red | Blue | Green | Yellow |
|---|---|---|---|---|
| Probability | 0.3 | 0.25 | 0.15 | x |
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of all the possible outcomes add up to 1. Add the ones we know, then take the total away from 1.
- Write down what we know. The outcomes listed cover every possibility, so their probabilities add up to 1. One of them is missing.P(red) = 0.3
P(blue) = 0.25
P(green) = 0.15
P(yellow) = ? - Add the known probabilities. Line up the decimal points carefully.0.3 + 0.25 + 0.15 = 0.7
- Subtract the total from 1. All the probabilities add up to 1, so the missing one is 1 minus the total of the others.1 − 0.7 = 0.3
Check. Add all the probabilities, including the one we found: 0.3 + 0.25 + 0.15 + 0.3 = 1. ✓
Watch out. Do not forget that the missing probability makes the total exactly 1 (not more, not less).
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of all the possible outcomes add up to 1. Add the ones we know, then take the total away from 1.
- Write down what we know. The outcomes listed cover every possibility, so their probabilities add up to 1. One of them is missing.P(win) = 12
P(draw) = 16
P(lose) = ? - Add the known probabilities. Use the common denominator 6 so that we can add them.36 + 16 = 46 = 23
- Subtract the total from 1. All the probabilities add up to 1, so the missing one is 1 minus the total of the others.66 − 46 = 26 = 13
Check. Add all the probabilities, including the one we found: 12 + 16 + 13 = 1. ✓
Watch out. Do not forget that the missing probability makes the total exactly 1 (not more, not less).
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. Red and blue cannot both happen on one spin, so the events are mutually exclusive. For mutually exclusive events we add the probabilities.
- Decide whether the events can happen together. The spinner lands on one colour only, so red and blue cannot both happen.Mutually exclusive
- Write the rule. For mutually exclusive events, P(A or B) = P(A) + P(B).P(red or blue) = P(red) + P(blue)
- Add the probabilities. Line up the decimal points.0.3 + 0.25 = 0.55
Check. The opposite event is green or yellow: 0.15 + 0.3 = 0.45 (since x = 0.3), and 0.55 + 0.45 = 1. ✓
Watch out. Add for “or” only when the events cannot happen together.
Watch out
Do not stop after finding the total of the known probabilities. The missing probability is 1 minus that total. And check that every probability is between 0 and 1 and that all of them add up to exactly 1. Also do not say P(not A) = P(A) − 1: that is negative.Exercise 23B Adding to 1 and the probability of “not”
Non-calculatorKey words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “A” happens or “not A” happens, never both, so these two events are opposites that cover every possibility.P(A) = 27
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not A) = 1 − P(A)
- Write 1 as a fraction with the same denominator. We can only subtract fractions with the same denominator. 1 whole is 7 out of 7.1 = 77
- Subtract the numerators. The denominator stays the same.77 − 27 = 57
Check. Add the two probabilities: 27 + 57 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “A” happens or “not A” happens, never both, so these two events are opposites that cover every possibility.P(A) = 59
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not A) = 1 − P(A)
- Write 1 as a fraction with the same denominator. We can only subtract fractions with the same denominator. 1 whole is 9 out of 9.1 = 99
- Subtract the numerators. The denominator stays the same.99 − 59 = 49
Check. Add the two probabilities: 59 + 49 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “A” happens or “not A” happens, never both, so these two events are opposites that cover every possibility.P(A) = 310
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not A) = 1 − P(A)
- Write 1 as a fraction with the same denominator. We can only subtract fractions with the same denominator. 1 whole is 10 out of 10.1 = 1010
- Subtract the numerators. The denominator stays the same.1010 − 310 = 710
Check. Add the two probabilities: 310 + 710 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “A” happens or “not A” happens, never both, so these two events are opposites that cover every possibility.P(A) = 712
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not A) = 1 − P(A)
- Write 1 as a fraction with the same denominator. We can only subtract fractions with the same denominator. 1 whole is 12 out of 12.1 = 1212
- Subtract the numerators. The denominator stays the same.1212 − 712 = 512
Check. Add the two probabilities: 712 + 512 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “A” happens or “not A” happens, never both, so these two events are opposites that cover every possibility.P(A) = 120
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not A) = 1 − P(A)
- Write 1 as a fraction with the same denominator. We can only subtract fractions with the same denominator. 1 whole is 20 out of 20.1 = 2020
- Subtract the numerators. The denominator stays the same.2020 − 120 = 1920
Check. Add the two probabilities: 120 + 1920 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “rain” happens or “no rain” happens, never both, so these two events are opposites that cover every possibility.P(rain) = 0.7
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(no rain) = 1 − P(rain)
- Subtract from 1. Line up the decimal points and take the probability away from 1.1 − 0.7 = 0.3
Check. Add the two probabilities: 0.7 + 0.3 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “win” happens or “not win” happens, never both, so these two events are opposites that cover every possibility.P(win) = 0.45
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not win) = 1 − P(win)
- Subtract from 1. Line up the decimal points and take the probability away from 1.1 − 0.45 = 0.55
Check. Add the two probabilities: 0.45 + 0.55 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “late” happens or “not late” happens, never both, so these two events are opposites that cover every possibility.P(late) = 0.08
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not late) = 1 − P(late)
- Subtract from 1. Line up the decimal points and take the probability away from 1.1 − 0.08 = 0.92
Check. Add the two probabilities: 0.08 + 0.92 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “pass” happens or “fail” happens, never both, so these two events are opposites that cover every possibility.P(pass) = 85%
- Write the rule. An event either happens or it does not, so the two percentages add up to 100%. Therefore P(not A) = 100% − P(A).P(fail) = 100% − P(pass)
- Subtract from 100%. Take the percentage away from 100.100% − 85% = 15%
Check. Add the two probabilities: 85% + 15% = 100%. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “sunny” happens or “not sunny” happens, never both, so these two events are opposites that cover every possibility.P(sunny) = 62%
- Write the rule. An event either happens or it does not, so the two percentages add up to 100%. Therefore P(not A) = 100% − P(A).P(not sunny) = 100% − P(sunny)
- Subtract from 100%. Take the percentage away from 100.100% − 62% = 38%
Check. Add the two probabilities: 62% + 38% = 100%. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “on time” happens or “not on time” happens, never both, so these two events are opposites that cover every possibility.P(on time) = 0.9
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not on time) = 1 − P(on time)
- Subtract from 1. Line up the decimal points and take the probability away from 1.1 − 0.9 = 0.1
Check. Add the two probabilities: 0.9 + 0.1 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “score” happens or “miss” happens, never both, so these two events are opposites that cover every possibility.P(score) = 45
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(miss) = 1 − P(score)
- Write 1 as a fraction with the same denominator. We can only subtract fractions with the same denominator. 1 whole is 5 out of 5.1 = 55
- Subtract the numerators. The denominator stays the same.55 − 45 = 15
Check. Add the two probabilities: 45 + 15 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
| Colour | Red | Blue | Green | Yellow |
|---|---|---|---|---|
| Probability | 0.2 | 0.35 | 0.15 | ? |
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of all the possible outcomes add up to 1. Add the ones we know, then take the total away from 1.
- Write down what we know. The outcomes listed cover every possibility, so their probabilities add up to 1. One of them is missing.P(red) = 0.2
P(blue) = 0.35
P(green) = 0.15
P(yellow) = ? - Add the known probabilities. Line up the decimal points carefully.0.2 + 0.35 + 0.15 = 0.7
- Subtract the total from 1. All the probabilities add up to 1, so the missing one is 1 minus the total of the others.1 − 0.7 = 0.3
Check. Add all the probabilities, including the one we found: 0.2 + 0.35 + 0.15 + 0.3 = 1. ✓
Watch out. Do not forget that the missing probability makes the total exactly 1 (not more, not less).
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of all the possible outcomes add up to 1. Add the ones we know, then take the total away from 1.
- Write down what we know. The outcomes listed cover every possibility, so their probabilities add up to 1. One of them is missing.P(red) = 310
P(blue) = 15
P(green) = ? - Add the known probabilities. Use the common denominator 10 so that we can add them.310 + 210 = 510 = 12
- Subtract the total from 1. All the probabilities add up to 1, so the missing one is 1 minus the total of the others.1010 − 510 = 510 = 12
Check. Add all the probabilities, including the one we found: 310 + 15 + 12 = 1. ✓
Watch out. Do not forget that the missing probability makes the total exactly 1 (not more, not less).
| Score | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Probability | 0.1 | 0.15 | 0.2 | 0.2 | 0.15 | ? |
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of all the possible outcomes add up to 1. Add the ones we know, then take the total away from 1.
- Write down what we know. The outcomes listed cover every possibility, so their probabilities add up to 1. One of them is missing.P(1) = 0.1
P(2) = 0.15
P(3) = 0.2
P(4) = 0.2
P(5) = 0.15
P(6) = ? - Add the known probabilities. Line up the decimal points carefully.0.1 + 0.15 + 0.2 + 0.2 + 0.15 = 0.8
- Subtract the total from 1. All the probabilities add up to 1, so the missing one is 1 minus the total of the others.1 − 0.8 = 0.2
Check. Add all the probabilities, including the one we found: 0.1 + 0.15 + 0.2 + 0.2 + 0.15 + 0.2 = 1. ✓
Watch out. Do not forget that the missing probability makes the total exactly 1 (not more, not less).
Method. The even scores are 2, 4 and 6. They cannot happen together, so add their probabilities.
- Find the even scores and their probabilities. The even numbers on a die are 2, 4 and 6.P(2) = 0.15, P(4) = 0.2, P(6) = 0.2
- Add the probabilities. Scores cannot happen together (mutually exclusive), so we add for “or”.0.15 + 0.2 + 0.2 = 0.55
Check. The odd scores are 1, 3 and 5: 0.1 + 0.2 + 0.15 = 0.45, and 0.55 + 0.45 = 1. ✓
Watch out. Use the probability for 6 you found in part (a).
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “a 6” happens or “not a 6” happens, never both, so these two events are opposites that cover every possibility.P(a 6) = 0.2
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(not a 6) = 1 − P(a 6)
- Subtract from 1. Line up the decimal points and take the probability away from 1.1 − 0.2 = 0.8
Check. Add the two probabilities: 0.2 + 0.8 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Method. If these were all the outcomes, their probabilities would add up to 1. Check whether they do.
- Add the three probabilities. Exhaustive outcomes add to 1.0.1 + 0.6 + 0.2 = 0.9
- Compare with 1. The total is less than 1, so the three outcomes do not cover every possibility. Something else can happen (for example, the train is cancelled).0.9 < 1
- Find the missing probability. Subtract the total from 1.1 − 0.9 = 0.1
Check. 0.1 + 0.6 + 0.2 + 0.1 = 1. ✓
Watch out. Missing probability = 1 minus the total of the known ones.
Method. Probabilities of all the outcomes must add up to exactly 1. Add Ali’s numbers and compare.
- Add the probabilities. These are all the possible outcomes, so they must total 1.0.5 + 0.3 + 0.4 = 1.2
- Compare with 1. A total over 1 is impossible.1.2 > 1
- Write the conclusion. Ali cannot be correct because the probabilities add up to more than 1.
Check. If one of them were 0.2 instead of 0.4 the total would be exactly 1. ✓
Watch out. A total bigger than 1 or smaller than 1 shows an error or a missing outcome.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “Ravi passes” happens or “Ravi fails” happens, never both, so these two events are opposites that cover every possibility.P(Ravi passes) = 0.58
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(Ravi fails) = 1 − P(Ravi passes)
- Subtract from 1. Line up the decimal points and take the probability away from 1.1 − 0.58 = 0.42
Check. Add the two probabilities: 0.58 + 0.42 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of an event happening and not happening add up to 1 (or 100%), so subtract the given probability from 1.
- Spot the event and its opposite. Either “Aisha passes” happens or “Aisha fails” happens, never both, so these two events are opposites that cover every possibility.P(Aisha passes) = 0.72
- Write the rule. An event either happens or it does not, so the two probabilities add up to 1. Therefore P(not A) = 1 − P(A).P(Aisha fails) = 1 − P(Aisha passes)
- Subtract from 1. Line up the decimal points and take the probability away from 1.1 − 0.72 = 0.28
Check. Add the two probabilities: 0.72 + 0.28 = 1. ✓
Watch out. The answer must be between 0 and 1. If you get a negative number or a number bigger than 1 you have subtracted the wrong way round.
Method. Find P(green) using the fact that the probabilities add up to 1, then change the probability into a number of sections.
- Add P(red) and P(blue). Use the common denominator 12.13 + 14 = 412 + 312 = 712
- Find P(green). Subtract the total from 1.1212 − 712 = 512
- Change to a number of sections. The bottom 12 is the number of sections, so the top is the number of green sections.5 sections
Check. Red is 12 ÷ 3 = 4 sections, blue is 12 ÷ 4 = 3 sections, and 4 + 3 + 5 = 12. ✓
Watch out. Do not forget to use the common denominator before adding thirds and quarters.
Key words. The complement of an event is “the event does not happen”, written not A or A′. Outcomes are exhaustive if together they cover every possibility, so their probabilities add up to 1.
Method. The probabilities of all the possible outcomes add up to 1. Add the ones we know, then take the total away from 1.
- Write down what we know. The outcomes listed cover every possibility, so their probabilities add up to 1. One of them is missing.P(red) = 0.4
P(blue) = 0.35
P(yellow) = ? - Add the known probabilities. Line up the decimal points carefully.0.4 + 0.35 = 0.75
- Subtract the total from 1. All the probabilities add up to 1, so the missing one is 1 minus the total of the others.1 − 0.75 = 0.25
Check. Add all the probabilities, including the one we found: 0.4 + 0.35 + 0.25 = 1. ✓
Watch out. Do not forget that the missing probability makes the total exactly 1 (not more, not less).
Method. Red and yellow cannot both be picked, so add their probabilities.
- Use the probability of yellow from part (a). P(yellow) = 0.25.P(yellow) = 0.25
- Add P(red) and P(yellow). Mutually exclusive events: add for “or”.0.4 + 0.25 = 0.65
Check. P(not blue) = 1 − 0.35 = 0.65. ✓
Watch out. Red or yellow is the same as not blue.
Method. Mutually exclusive events cannot happen together, so P(A or B) = P(A) + P(B). A probability cannot be bigger than 1.
- Suppose they were mutually exclusive. Then we could add the probabilities to get P(A or B).P(A or B) = 0.7 + 0.5 = 1.2
- Compare with the largest possible probability. P(A or B) must be at most 1.1.2 > 1, which is impossible
- Conclude. So the assumption was wrong.A and B can happen together, so they are not mutually exclusive.
Check. Two events with probabilities 0.4 and 0.5 could be mutually exclusive (0.4 + 0.5 = 0.9). ✓
Watch out. Mutually exclusive events must have probabilities that add up to 1 or less.