Exactly one of two events

We are often interested in determining when exactly one of two events has occurred, but not both. For example, let $A$ be the event "a student passes the first exam" and $B$ the event "the student passes the second exam". The student passes exactly one exam by passing the first and failing the second, or by failing the first and passing the second.

The first case is $A \cap B^c$ and the second is $A^c \cap B$. Either case counts, so we take the union of the two events.

This is not $A \cup B$, because the union also contains the overlap $A \cap B$, where both events occur.

The diagram below builds the event on a Venn diagram in four steps. Work through them with the buttons.

Two coins are flipped, so $\Omega = \{HH, HT, TH, TT\}$. Let $A$ be the event "the first coin shows heads" and $B$ the event "the second coin shows heads". Write the event "exactly one head" as a set expression in $A$ and $B$, and list its outcomes.

Solution

Exactly one head means one coin shows heads and the other does not. Either the first coin shows heads and the second does not, which is $A \cap B^c$, or the second shows heads and the first does not, which is $A^c \cap B$. Either case counts, so the event is the union:

To list it, we write out the two events and their complements:

The only outcome in both $A$ and $B^c$ is $HT$, and the only outcome in both $A^c$ and $B$ is $TH$:

These are the differences $A \setminus B$ and $B \setminus A$. Joining them gives the event:

The outcome $HH$ lies in the overlap, where both events occur, and $TT$ lies outside both circles, where neither occurs. Neither is in the shaded regions below.

Another way to build "exactly one of $A$, $B$" starts from the union. The union $A \cup B$ is the event that at least one of $A$, $B$ occurs. It contains every outcome where exactly one occurs, but also the overlap $A \cap B$, where both occur.

So we remove the overlap from the union. This is the set difference

The diagram below builds the event in two steps. Work through them with the buttons.

Writing the set difference as an intersection with a complement gives a second form.

On the diagram, removing the overlap from the union leaves the outcomes in $A$ only, $A \cap B^c$, and those in $B$ only, $A^c \cap B$. So both forms describe the same event as $(A \cap B^c) \cup (A^c \cap B)$.

All three describe the same event. Many events can be written in more than one way, and which form is easiest to work with depends on the information we have.

Here, if $A$ and $B$ are listed, the first applies directly. If we know only the union and the overlap, the second does. If we know only that at least one occurs and that not both occur, the third does.

A spinner lands on one of the numbers $1$ to $8$, so $\Omega = \{1, 2, \ldots, 8\}$. Let $A$ be "the number is at most $5$" and $B$ be "the number is odd". Find the event that exactly one of $A$, $B$ occurs.

Solution

Listing the outcomes of each event:

The union holds the numbers in at least one event, and the overlap holds the numbers in both:

Exactly one event occurs on the union with the overlap removed:

The numbers $2$ and $4$ are at most $5$ but even, and $7$ is odd but greater than $5$.