Mutually Exclusive Events

Roll one six-sided die. "The roll is even" and "the roll is a 5" can't both happen on that roll, because 5 is odd. "The roll is even" and "the roll is greater than 3" can both happen: a 4 or a 6 does it. Whether two events can happen together is a probability question, and the probability that answers it has a name.

Joint probability: both events at once

For two events $A$ and $B$ in the same random process, the event "$A$ and $B$" is the set of outcomes that belong to both. It is called the intersection of $A$ and $B$ and written $A \cap B$. Its probability, $P(A \cap B)$, is the joint probability of $A$ and $B$: the chance that one run of the process produces an outcome in both events.

With the die, let $A$ = even $= \{2, 4, 6\}$ and $C$ = greater than 3 $= \{4, 5, 6\}$. The outcomes in both are $\{4, 6\}$, and the six outcomes are equally likely (topic 2.4), so

The joint probability is never larger than either event's own probability, because $A \cap C$ is part of $A$ and also part of $C$. Here $\frac{1}{3}$ is less than $P(A) = \frac{1}{2}$ and less than $P(C) = \frac{1}{2}$.

Reading a joint probability from a two-way table

Most AP questions give the joint probability as a cell of a two-way table. A public library recorded the 500 items checked out on one Saturday by format and by the borrower's age group:

Pick one of these 500 checkouts at random. The event "ebook and child" is the one cell where the Ebook row meets the Child column, so

The denominator is the grand total, 500, because the random selection is from all 500 checkouts. Two nearby numbers are different questions: $\frac{140}{500} = 0.28$ is $P(\text{ebook})$, which ignores age, and $\frac{30}{180}$ is the proportion of children's checkouts that were ebooks, a conditional probability (topic 2.6).

Mutually exclusive events

Two events are mutually exclusive, also called disjoint, when they cannot both occur on the same run of the process. They share no outcomes, so their intersection is empty and

The reverse also holds: if the joint probability is 0, the events never happen together, so they are mutually exclusive. That makes the joint probability the tool for justifying either answer.

In the library table, every checkout has exactly one format and exactly one age group. So "print book" and "ebook" are mutually exclusive, and so are "child" and "adult": the categories of one variable never overlap when each individual gets one value. Events built from two different variables are mutually exclusive only if their cell is 0. "Ebook" and "child" are not, because $P(\text{ebook} \cap \text{child}) = 0.06 > 0$.

What a zero cell does and doesn't show

A zero in a table of data shows that among these individuals no one fell in both categories, so for a random pick from them the joint probability is exactly 0. For a larger population, it means you saw no overlap in this sample, which is evidence but not proof. When a question asks about the individuals in the table, the zero cell settles it. When the events can't overlap by their definitions (one die roll can't be both 2 and 5), they are mutually exclusive for every run, with or without data.

A quick test with two probabilities

If $A$ and $B$ are mutually exclusive, their outcomes sit in separate parts of the sample space, so together they use up $P(A) + P(B)$ of the total probability, which is 1. That gives a fast check: if $P(A) + P(B) > 1$, the events cannot be mutually exclusive. With $P(A) = 0.6$ and $P(B) = 0.5$, the two events would need 1.1 of the probability if they didn't overlap, and only 1 exists, so at least $0.1$ of the probability must be in both.

The test only works in one direction. $P(A) + P(B) \le 1$ does not show the events are mutually exclusive; it only fails to rule it out. The die events "even" and "greater than 3" have $\frac{1}{2} + \frac{1}{2} = 1$ and still overlap.

Mutually exclusive is a different idea from independent. If $A$ and $B$ are disjoint and you learn $A$ happened, you know for certain that $B$ did not, which is the opposite of the two events having nothing to do with each other. Topic 2.7 defines independence and shows why disjoint events with positive probabilities are never independent.

Worked examples

Where students lose points