GCSE Maths: Independent events: the AND rule and the OR rule (Foundation)
Part of the GCSE Maths probability chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Independent events: the AND rule and the OR rule
Two events are independent if the result of one has no effect on the result of the other. Tossing a coin and rolling a die are independent: whatever the coin shows, the die is still fair. Rain on two different, far-apart days is usually treated as independent too. In real life we assume independence only when it is reasonable.
The AND rule. To find the probability that both of two independent events happen, multiply the probabilities: P(A and B) = P(A) × P(B). Why? Look at the table of the 12 outcomes when a coin is tossed and a die is rolled. Only one cell, H6, is “heads and a 6”, so the probability is 1/12. And 1/2 × 1/6 = 1/12. Half of all the outcomes have heads, and one sixth of those have a 6, so we take one sixth of one half, which is multiplying.
| coin \ die | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| H | H1 | H2 | H3 | H4 | H5 | H6 |
| T | T1 | T2 | T3 | T4 | T5 | T6 |
The OR rule. Two events are mutually exclusive if they cannot happen at the same time (a die cannot show 2 and 5 on the same roll). For such events, the probability that either happens is found by adding: P(A or B) = P(A) + P(B). For example P(roll 2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.
Do not mix them up. “And” (both things, one after the other) means multiply, and the answer is smaller. “Or” (one thing or the other, in the same trial) means add, and the answer is bigger. The AND rule needs independent events. The OR rule needs mutually exclusive events.
At least one. “At least one 6 in two rolls” means one 6, or two sixes. That is lots of cases, so use the opposite instead: P(at least one) = 1 − P(none). P(no 6 on one roll) = 5/6, so P(no 6 in two rolls) = 5/6 × 5/6 = 25/36, and P(at least one 6) = 1 − 25/36 = 11/36. Notice that this is not 1/6 + 1/6 = 1/3: that adds events that are not mutually exclusive (both rolls could be sixes).
Key ideas
- Independent events do not affect each other. The AND rule: P(A and B) = P(A) × P(B).
- Mutually exclusive events cannot happen together. The OR rule: P(A or B) = P(A) + P(B).
- “And” means multiply (smaller answer). “Or” means add (bigger answer).
- P(at least one) = 1 − P(none).
- We should say what we assume: for example, that the coin is fair and the events are independent.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(heads) = 12
P(a 6) = 16 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply the tops together and the bottoms together.12 × 16 = 112
Check. Both things have to happen, so the answer must be smaller than each single probability: 112 is smaller than 16. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. Rain on one day does not change the chance of rain on the other (we assume this). So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(rain on Saturday) = 0.3
P(rain on Sunday) = 0.4 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply as if there were no decimal points, then count the decimal places.0.3 × 0.4
3 × 4 = 12, and there are 2 decimal places altogether
= 0.12
Check. Both things have to happen, so the answer must be smaller than each single probability: 0.12 is smaller than 0.3. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For mutually exclusive events, “A or B” means we add their probabilities: P(A or B) = P(A) + P(B).
- Check that the events are mutually exclusive. One counter cannot be both red and blue. So we can add their probabilities.Mutually exclusive events
- Write down the probability of each event. We need each probability on its own.P(red) = 310
P(blue) = 15 - Add the probabilities. For mutually exclusive events, P(A or B) = P(A) + P(B).310 + 15 = 310 + 210 = 510 = 12
Check. The opposite event has probability 1 − 12 = 12, and the two add to 1. ✓
Watch out. Add for “or”, but only when the events cannot both happen.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. P(at least one) = 1 − P(none). Find the probability of none by multiplying, then subtract from 1.
- Use the opposite event. “At least one 6” is the opposite of “no 6 every time”. It is easier to find the opposite and subtract from 1.P(at least one) = 1 − P(none)
- Find the probability of the event not happening on one trial. Use P(not A) = 1 − P(A).P(not 6) = 1 − 16 = 56
- Multiply for all 2 trials. The trials are independent, so multiply the 2 probabilities.P(none) = 56 × 56 = 2536
- Subtract from 1. P(at least one) = 1 − P(none).3636 − 2536 = 1136
Check. P(none) + P(at least one) = 2536 + 1136 = 1. ✓
Watch out. Do not add the probabilities of each trial. For “at least one” always think “1 minus none”.
Watch out
Do not add when you should multiply. The probability of heads and a 6 is 1/2 × 1/6 = 1/12, not 1/2 + 1/6. And do not use the OR rule on events that can both happen: “a 6 on the first roll or a 6 on the second roll” is not 1/6 + 1/6, because both can be sixes.Exercise 23F Independent events: AND and OR
Non-calculatorKey words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(heads) = 12
P(a 3) = 16 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply the tops together and the bottoms together.12 × 16 = 112
Check. Both things have to happen, so the answer must be smaller than each single probability: 112 is smaller than 16. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(tails) = 12
P(an even number) = 12 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply the tops together and the bottoms together.12 × 12 = 14
Check. Both things have to happen, so the answer must be smaller than each single probability: 14 is smaller than 12. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(heads) = 12
P(a number bigger than 4) = 13 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply the tops together and the bottoms together.12 × 13 = 16
Check. Both things have to happen, so the answer must be smaller than each single probability: 16 is smaller than 13. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(first coin heads) = 12
P(second coin heads) = 12 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply the tops together and the bottoms together.12 × 12 = 14
Check. Both things have to happen, so the answer must be smaller than each single probability: 14 is smaller than 12. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(a 6 first) = 16
P(a 6 second) = 16 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply the tops together and the bottoms together.16 × 16 = 136
Check. Both things have to happen, so the answer must be smaller than each single probability: 136 is smaller than 16. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(A) = 0.3
P(B) = 0.5 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply as if there were no decimal points, then count the decimal places.0.3 × 0.5
3 × 5 = 15, and there are 2 decimal places altogether
= 0.15
Check. Both things have to happen, so the answer must be smaller than each single probability: 0.15 is smaller than 0.3. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(A) = 0.6
P(B) = 0.2 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply as if there were no decimal points, then count the decimal places.0.6 × 0.2
6 × 2 = 12, and there are 2 decimal places altogether
= 0.12
Check. Both things have to happen, so the answer must be smaller than each single probability: 0.12 is smaller than 0.2. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(A) = 0.9
P(B) = 0.9 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply as if there were no decimal points, then count the decimal places.0.9 × 0.9
9 × 9 = 81, and there are 2 decimal places altogether
= 0.81
Check. Both things have to happen, so the answer must be smaller than each single probability: 0.81 is smaller than 0.9. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(A) = 0.04
P(B) = 0.5 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply as if there were no decimal points, then count the decimal places.0.04 × 0.5
4 × 5 = 20, and there are 3 decimal places altogether
= 0.02
Check. Both things have to happen, so the answer must be smaller than each single probability: 0.02 is smaller than 0.04. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For mutually exclusive events, “A or B” means we add their probabilities: P(A or B) = P(A) + P(B).
- Check that the events are mutually exclusive. These events cannot happen at the same time. So we can add their probabilities.Mutually exclusive events
- Write down the probability of each event. We need each probability on its own.P(A) = 0.3
P(B) = 0.5 - Add the probabilities. For mutually exclusive events, P(A or B) = P(A) + P(B).0.3 + 0.5 = 0.8
Check. The opposite event has probability 1 − 0.8 = 0.2, and the two add to 1. ✓
Watch out. Add for “or”, but only when the events cannot both happen.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For mutually exclusive events, “A or B” means we add their probabilities: P(A or B) = P(A) + P(B).
- Check that the events are mutually exclusive. These events cannot happen at the same time. So we can add their probabilities.Mutually exclusive events
- Write down the probability of each event. We need each probability on its own.P(A) = 0.15
P(B) = 0.25 - Add the probabilities. For mutually exclusive events, P(A or B) = P(A) + P(B).0.15 + 0.25 = 0.4
Check. The opposite event has probability 1 − 0.4 = 0.6, and the two add to 1. ✓
Watch out. Add for “or”, but only when the events cannot both happen.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For mutually exclusive events, “A or B” means we add their probabilities: P(A or B) = P(A) + P(B).
- Check that the events are mutually exclusive. These events cannot happen at the same time. So we can add their probabilities.Mutually exclusive events
- Write down the probability of each event. We need each probability on its own.P(A) = 0.07
P(B) = 0.4 - Add the probabilities. For mutually exclusive events, P(A or B) = P(A) + P(B).0.07 + 0.4 = 0.47
Check. The opposite event has probability 1 − 0.47 = 0.53, and the two add to 1. ✓
Watch out. Add for “or”, but only when the events cannot both happen.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For mutually exclusive events, “A or B” means we add their probabilities: P(A or B) = P(A) + P(B).
- Check that the events are mutually exclusive. The spinner cannot land on two colours at once. So we can add their probabilities.Mutually exclusive events
- Write down the probability of each event. We need each probability on its own.P(red) = 0.2
P(blue) = 0.3 - Add the probabilities. For mutually exclusive events, P(A or B) = P(A) + P(B).0.2 + 0.3 = 0.5
Check. The opposite event has probability 1 − 0.5 = 0.5, and the two add to 1. ✓
Watch out. Add for “or”, but only when the events cannot both happen.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For mutually exclusive events, “A or B” means we add their probabilities: P(A or B) = P(A) + P(B).
- Check that the events are mutually exclusive. These events cannot happen at the same time. So we can add their probabilities.Mutually exclusive events
- Write down the probability of each event. We need each probability on its own.P(1) = 16
P(6) = 16 - Add the probabilities. For mutually exclusive events, P(A or B) = P(A) + P(B).16 + 16 = 26 = 13
Check. The opposite event has probability 1 − 13 = 23, and the two add to 1. ✓
Watch out. Add for “or”, but only when the events cannot both happen.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For mutually exclusive events, “A or B” means we add their probabilities: P(A or B) = P(A) + P(B).
- Check that the events are mutually exclusive. These events cannot happen at the same time. So we can add their probabilities.Mutually exclusive events
- Write down the probability of each event. We need each probability on its own.P(2) = 16
P(3) = 16
P(4) = 16 - Add the probabilities. For mutually exclusive events, P(A or B) = P(A) + P(B).16 + 16 + 16 = 36 = 12
Check. The opposite event has probability 1 − 12 = 12, and the two add to 1. ✓
Watch out. Add for “or”, but only when the events cannot both happen.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(bus late) = 0.2
P(train late) = 0.1 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply as if there were no decimal points, then count the decimal places.0.2 × 0.1
2 × 1 = 2, and there are 2 decimal places altogether
= 0.02
Check. Both things have to happen, so the answer must be smaller than each single probability: 0.02 is smaller than 0.1. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. P(bus not late) = 1 − 0.2 = 0.8 and P(train not late) = 1 − 0.1 = 0.9. They are independent. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(bus not late) = 0.8
P(train not late) = 0.9 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply as if there were no decimal points, then count the decimal places.0.8 × 0.9
8 × 9 = 72, and there are 2 decimal places altogether
= 0.72
Check. Both things have to happen, so the answer must be smaller than each single probability: 0.72 is smaller than 0.8. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Method. “At least one late” is the opposite of “neither late”. Use the answer to part (b).
- Recall P(neither late). From part (b).0.8 × 0.9 = 0.72
- Subtract from 1. P(at least one late) = 1 − P(neither late).1 − 0.72 = 0.28
Check. Both late is 0.02, bus only 0.18, train only 0.08: 0.02 + 0.18 + 0.08 = 0.28. ✓
Watch out. Use 1 minus the opposite.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. P(at least one) = 1 − P(none). Find the probability of none by multiplying, then subtract from 1.
- Use the opposite event. “At least one head” is the opposite of “no heads every time”. It is easier to find the opposite and subtract from 1.P(at least one) = 1 − P(none)
- Find the probability of the event not happening on one trial. Use P(not A) = 1 − P(A).P(not head) = 1 − 12 = 12
- Multiply for all 2 trials. The trials are independent, so multiply the 2 probabilities.P(none) = 12 × 12 = 14
- Subtract from 1. P(at least one) = 1 − P(none).44 − 14 = 34
Check. P(none) + P(at least one) = 14 + 34 = 1. ✓
Watch out. Do not add the probabilities of each trial. For “at least one” always think “1 minus none”.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. P(at least one) = 1 − P(none). Find the probability of none by multiplying, then subtract from 1.
- Use the opposite event. “At least one head” is the opposite of “no heads every time”. It is easier to find the opposite and subtract from 1.P(at least one) = 1 − P(none)
- Find the probability of the event not happening on one trial. Use P(not A) = 1 − P(A).P(not head) = 1 − 12 = 12
- Multiply for all 3 trials. The trials are independent, so multiply the 3 probabilities.P(none) = 12 × 12 × 12 = 18
- Subtract from 1. P(at least one) = 1 − P(none).88 − 18 = 78
Check. P(none) + P(at least one) = 18 + 78 = 1. ✓
Watch out. Do not add the probabilities of each trial. For “at least one” always think “1 minus none”.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. P(at least one) = 1 − P(none). Find the probability of none by multiplying, then subtract from 1.
- Use the opposite event. “At least one 6” is the opposite of “no 6 every time”. It is easier to find the opposite and subtract from 1.P(at least one) = 1 − P(none)
- Find the probability of the event not happening on one trial. Use P(not A) = 1 − P(A).P(not 6) = 1 − 16 = 56
- Multiply for all 2 trials. The trials are independent, so multiply the 2 probabilities.P(none) = 56 × 56 = 2536
- Subtract from 1. P(at least one) = 1 − P(none).3636 − 2536 = 1136
Check. P(none) + P(at least one) = 2536 + 1136 = 1. ✓
Watch out. Do not add the probabilities of each trial. For “at least one” always think “1 minus none”.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. For independent events, “A and B” means we multiply their probabilities: P(A and B) = P(A) × P(B).
- Check that the events are independent. The outcome of one does not change the chances for the other. So we can multiply their probabilities.Independent events
- Write down the probability of each event. We need each probability on its own.P(a number bigger than 3) = 25
P(tails) = 12 - Multiply the probabilities. For independent events, P(A and B) = P(A) × P(B). Multiply the tops together and the bottoms together.25 × 12 = 210 = 15
Check. Both things have to happen, so the answer must be smaller than each single probability: 15 is smaller than 25. ✓
Watch out. Multiply for “and”. Do not add the probabilities.
Method. Exactly one germinates in two ways: the first does and the second does not, or the first does not and the second does. Find each way, then add.
- Find P(not germinate). Use P(not A) = 1 − P(A).1 − 0.8 = 0.2
- Way 1: first germinates, second does not. Independent events, so multiply.0.8 × 0.2 = 0.16
- Way 2: first does not, second does. Multiply again.0.2 × 0.8 = 0.16
- Add the two ways. They are different ways (this or that), so we add.0.16 + 0.16 = 0.32
Check. Both germinate: 0.64. Neither: 0.04. So 0.64 + 0.32 + 0.04 = 1. ✓
Watch out. “Exactly one” has two orders. Do not forget the second way.
Method. “Both” means “and”, and for independent events “and” means multiply.
- Read the question. We want a 6 on the first roll and a 6 on the second roll.“and”
- Say what Ali did. He added, but adding is for “or” with events that cannot happen together.Ali added: wrong
- Do it correctly. Multiply the probabilities.16 × 16 = 136
Check. There are 36 outcomes for two dice and only (6, 6) works. ✓
Watch out. Both things happening is rarer than one, so the answer must be smaller than 1/6.
Method. Check whether the answer is sensible: “both” must be less likely than either single day.
- Compare. Rain on both days can only happen if it rains on Saturday, so its probability cannot be more than P(rain on Saturday) = 0.3.0.7 > 0.3, which is impossible
- Find the correct answer. Independent events: multiply.0.3 × 0.4 = 0.12
Check. 0.12 is smaller than both 0.3 and 0.4. ✓
Watch out. Use a quick size check: “and” answers are smaller than the single probabilities.
Method. Mutually exclusive events cannot happen at the same time. Check whether one outcome is in both events.
- List the outcomes of each event. A: 2, 4, 6. B: 6.A = {2, 4, 6}, B = {6}
- Look for a shared outcome. If the same outcome is in both, both can happen at once.6 is in both
- Conclude. Not mutually exclusive.
Check. If the die shows 6, both A and B have happened. ✓
Watch out. Check for a shared outcome.
Method. Mutually exclusive events cannot happen at the same time. Check whether one outcome is in both events.
- List the outcomes of each event. A: 2, 4, 6. B: 5.A = {2, 4, 6}, B = {5}
- Look for a shared outcome. There is no outcome in both lists.No shared outcome
- Conclude. They cannot happen together.Mutually exclusive.
Check. A score of 5 is odd, so it cannot also be even. ✓
Watch out. Check for a shared outcome.
Key words. Two events are independent if one happening does not change the probability of the other. Mutually exclusive events cannot happen at the same time. For independent events, P(A and B) = P(A) × P(B).
Method. P(at least one) = 1 − P(none). Find the probability of none by multiplying, then subtract from 1.
- Use the opposite event. “At least one 6” is the opposite of “no 6 every time”. It is easier to find the opposite and subtract from 1.P(at least one) = 1 − P(none)
- Find the probability of the event not happening on one trial. Use P(not A) = 1 − P(A).P(not 6) = 1 − 16 = 56
- Multiply for all 3 trials. The trials are independent, so multiply the 3 probabilities.P(none) = 56 × 56 × 56 = 125216
- Subtract from 1. P(at least one) = 1 − P(none).216216 − 125216 = 91216
Check. P(none) + P(at least one) = 125216 + 91216 = 1. ✓
Watch out. Do not add the probabilities of each trial. For “at least one” always think “1 minus none”.