Mutually Exclusive Events A and B

What Does Mutually Exclusive Mean?

Imagine you have two choices, but you can only pick one, not both at the same time. Well, in the world of probability, this is similar to the concept of Mutually Exclusive Events (other cool names: Disjoint Events).

Two events (let's call them event $A$ and event $B$) are said to be mutually exclusive if both events cannot possibly occur at the same time in a single trial. Simply put, if $A$ happens, $B$ cannot happen. If $B$ happens, $A$ cannot happen.

Characteristics of Mutually Exclusive Events

The main characteristic is just that: cannot happen simultaneously. There's no outcome that can belong to event $A$ and event $B$ at the same time.

Probability of Two Events Occurring Together

Because events $A$ and $B$ cannot happen together if they are mutually exclusive, the probability of both occurring simultaneously is zero!

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

Or using the intersection symbol:

Remember, if they are mutually exclusive, their intersection is empty, so the probability is zero!

Calculating the Combined Probability for Mutually Exclusive Events

So, how do we calculate the probability of event $A$ happening OR event $B$ happening if $A$ and $B$ are mutually exclusive?

Since they can't happen together, we simply add the probabilities of each individual event.

The formula becomes very easy:

Or using the union symbol:

This is the Special Addition Rule which applies ONLY to mutually exclusive events. (If the events are not mutually exclusive, there's a slightly different formula).

Examples of Mutually Exclusive Events

To understand better, look at these examples:

1. Coin Toss:

   The event of getting "Heads" and the event of getting "Tails". They can't happen together, right?

   • $P(\text{Heads}) = 1/2$
   • $P(\text{Tails}) = 1/2$
   • 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):

   Look at 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 $26/52 + 13/52 = 39/52 = 3/4$.