### What is a mutually exclusive event?

Mutually exclusive events are things that can’t happen at the same time. For instance, you can’t run forward and in reverse simultaneously. The actions “running forward” and “running in reverse” are mutually exclusive. Flipping a coin can also give you this sort of event. You can’t flip a coin and get both heads and tails. So “getting heads” and “getting tails” are mutually exclusive events. Some other examples include your capacity to pay your rent if you don’t get paid or to turn off the TV in the event that you don’t have a TV.

### Probability

The basic probability(P) of an event happening (forgetting mutual exclusivity for a moment) is:

P = number of ways the event can happen / total number of outcomes.

Example: The probability of rolling a 5 when you roll a die is 1/6 because there is one 5 on a die and there are six possible outcomes. If we call the probability of rolling a 5 “Event A”, then the equation is:

P(A) = number of ways the event can happen / total number of outcomes

P(A) = 1 / 6.

It’s impossible to roll a 5 and a 6 together; the events are mutually exclusive.

The events are written like this:

P(A and B) = 0

Likewise, In English, this means that the probability of event A (rolling a 5) and event B (rolling a 6) happening together is 0.

However, when you roll a die, you can roll a 5 OR a 6 (the odds are 1 out of 6 for each event) and the sum of either event happening is the sum of both probabilities. In probability, it’s written like this:

P(A or B) = P(A) + P(B)

P(rolling a 5 or rolling a 6) = P(rolling a 5) + P(rolling a 6)

P(rolling a 5 or rolling a 6) = 1/6 + 1/6 = 2/6 = 1/3.

## Mutual exclusivity

It’s impossible to roll a 1 and a 2 together.

**Example**

Sample problem: “If P(A) = 0.20, P(B) = 0.35 and (P AU B) = 0.51, are A and B mutually exclusive?”

Note: a union (U) of two events occurring means that A or B occurs.

**Step 1: **Add up the probabilities of the separate events (A and B). In the above example:

.20 + .35 = .55

**Step 2:** Compare the r answer to the given union statement (A U B). If they are the same, that means that the events are mutually exclusive. If they are not the same, they are not mutually exclusive. This is because if they are mutually exclusive (meaning that they can’t occur together), then the (U)nion of the two events must be the sum of both, i.e. 0.20 + 0.35 = 0.55.

In our example, 0.55 does not equal 0.51, so the events are not mutually exclusive.