Mutually Exclusive Events A and B

Events with No Shared Outcome

Suppose one trial can produce either of two outcomes, but not both. In probability, events with this relationship are mutually exclusive, or disjoint.

Events $A$ and $B$ are mutually exclusive when they cannot occur together in a single trial. If $A$ occurs, $B$ does not occur, and vice versa.

Characteristics of Mutually Exclusive Events

Their defining property is an empty intersection: no outcome belongs to both event $A$ and event $B$. That property is what allows the two probabilities to be added directly, because no outcome would be counted twice.

Probability of Two Events Occurring Together

Because events $A$ and $B$ cannot occur together, the probability of their intersection is zero.

We can write the probability of event "$A$ and $B$" (both occurring) as:

Or using the intersection symbol:

For mutually exclusive events, the intersection is empty, so its probability is zero.

Calculating the Combined Probability for Mutually Exclusive Events

To calculate the probability that event $A$ OR event $B$ occurs, use the fact that the events have no shared outcomes.

The two individual probabilities can therefore be added without double-counting.

The formula is:

Or using the union symbol:

This special addition rule applies only to mutually exclusive events. When events overlap, their intersection must also be included in the calculation.

Examples of Mutually Exclusive Events

The following examples show when the special addition rule applies:

1. Coin Toss:

   A single coin toss produces either "Heads" or "Tails," but not both.

   • $P(\text{Heads}) = 1/2$
   • $P(\text{Tails}) = 1/2$
   • Probability of getting Heads OR Tails is $P(\text{Heads}) + P(\text{Tails}) = 1/2 + 1/2 = 1$ (One of them must occur).

2. Rolling a Die (once):

   Consider two pairs of events:

   • Getting a $3$ and getting a $5$ cannot happen at once. Each has probability $1/6$, so the probability of getting a $3$ or $5$ is $1/6 + 1/6 = 2/6 = 1/3$.
   • Getting an even number $\{2, 4, 6\}$ and getting an odd number $\{1, 3, 5\}$ cannot happen at once. Each event has probability $3/6 = 1/2$, so the probability of getting even or odd is $1/2 + 1/2 = 1$.

3. Drawing a Card (once):

   Use the same idea for card events:

   • Getting a King and getting a Queen are mutually exclusive. There are $4$ Kings and $4$ Queens in $52 \text{ cards}$, so the probability of getting a King or Queen is $4/52 + 4/52 = 8/52 = 2/13$.
   • Getting a red card and getting a club $\clubsuit$ are mutually exclusive because clubs are black. There are $26$ red cards and $13$ clubs, so the probability of getting red or club is $26/52 + 13/52 = 39/52 = 3/4$.

Exercise

A fair six-sided die is rolled once. Let $A=\{1,2\}$ and $B=\{5,6\}$.

1. Show that $A$ and $B$ are mutually exclusive.
2. Find $P(A \cup B)$.
3. Find the probability that neither event occurs.

Worked Solution

The two events have an empty intersection and therefore no shared outcome:

They are therefore mutually exclusive. Each event contains $2$ of the $6$ equally likely die outcomes, so the special addition rule gives

The outcomes outside $A \cup B=\{1,2,5,6\}$ are $\{3,4\}$. Thus,

As a check, the probability of the union and the probability of its complement add to $2/3+1/3=1$.