Unit 2 · Topic 2.5 · about 15 minutes

Mutually Exclusive Events

Decide whether two events are mutually exclusive and justify the decision with their joint probability.

Predict first

Roll a fair die once. Event A is "roll a 1" and event B is "roll a 6". Now roll the die twice. Event C is "the first roll is a 1" and event D is "the second roll is a 6". Which pair of events cannot happen together?

Joint probability

The probability that events A and B both occur is their joint probability. It is the probability of the intersection of A and B, written $P (A \cap B)$ and read "the probability of A intersect B", or just "A and B".

In a two-way table a joint probability comes from one cell. The table below records 400 adults by their main way of getting to work and whether they ever work from home. If one of the 400 is chosen at random, the probability that the adult drives and works from home some days is $\frac{74}{400} = 0.185$ .

400 adults by main way of getting to work and working from home (counts)
Main way to work	Works from home some days	Never works from home	Total
Drives	74	146	220
Takes transit	28	72	100
Walks or bikes	18	62	80
Total	120	280	400

Mutually exclusive events

Two events are mutually exclusive, also called disjoint, if they cannot occur at the same time. No outcome belongs to both, so their joint probability is zero:

P (A \cap B) = 0

To justify that two events are mutually exclusive, show that their joint probability is 0. To show that they are not, point to an outcome that belongs to both, or to a joint probability that is not 0.

In the commute table, "drives" and "takes transit" are mutually exclusive. Each adult reported one main way of getting to work, so no adult sits in both rows, and $P (drives \cap transit) = 0$ . "Drives" and "works from home some days" are not mutually exclusive: 74 adults are in both, and $\frac{74}{400} = 0.185$ is not 0.

Sort it

Each card describes one trial and two events. Tap a card, then tap the bin it belongs in.

Mutually exclusive

Not mutually exclusive

Worked exampleCan these two groups be separate?

A streaming service reports that last week, of its 1,000 subscribers, 550 watched a comedy, 600 watched a drama, and 310 watched both. A marketing intern writes that comedy watchers and drama watchers are two separate groups. Use joint probability to show that the intern is wrong.

Define the events. For a randomly chosen subscriber, let C be "watched a comedy last week" and D be "watched a drama last week". Then $P (C) = 0.55$ and $P (D) = 0.60$ .
Find the joint probability. 310 subscribers watched both, so $P (C \cap D) = \frac{310}{1000} = 0.31$ .
Compare it with 0. Mutually exclusive events cannot happen together, so their joint probability is 0. Here it is 0.31.
Conclude in context. The joint probability is 0.31, not 0, so watching a comedy and watching a drama are not mutually exclusive. 31% of the subscribers watched both.

Answer.

The events are not mutually exclusive, because $P (C \cap D) = 0.31$ , not 0.

Check your understanding

A survey of 300 students found that 80 are in the band, 120 play a sport and 45 do both. One of these students is chosen at random. Which statement gives the correct decision about whether "in the band" and "plays a sport" are mutually exclusive, with a valid reason?

At a bakery, the probability that a randomly chosen customer buys bread is 0.40, and the probability that the customer buys a cake is 0.25. Using only these two probabilities, can you tell whether "buys bread" and "buys a cake" are mutually exclusive?

A whole number from 1 to 20 is chosen at random. Which pair of events is mutually exclusive?

A survey asks each adult to name one favorite season. For a randomly chosen adult, let A be "names summer" and B be "names winter". Which statement gives the correct conclusion with a valid reason?

Course alignment, for teachers

AP Statistics topic 2.5, Unit 2: Probability, Random Variables, and Probability Distributions.

Skill 4.B: Justify a claim based on statistical calculations and results.