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# Mutually Exclusive Events

### Mutually Exclusive Events

Events that cannot happen at the same time are known as mutually exclusive.

Mutually exclusive events are events that cannot happen at the same time. Which option below shows mutually exclusive events?

Consider these two events - are they mutually exclusive?

```
1. Rolling a 6
2. Not rolling a 6
```

The two events rolling a 6 and not rolling a 6 are special - they are mutually exclusive and exhaustive. What does this mean?

Mutually exclusive events are exhaustive if it is certain that one of them will happen.

Another example is: 1. Picking a yellow ball from this bag 2. Not picking a yellow ball from this bag

Exhaustive mutually exclusive events have a special property. We have seen that it is certain we will either roll a 6 or not roll a 6. On the probability scale, what number indicates the probability is certain?

This is a really important result and we can write it as follows:

$P(6)+P(not6)=1$.

This means we can easily find the probability of an event not happening. We know that $P(6)=\frac{1}{6}$, so what is $P(not6)$?

We noted that these two events are mutually exclusive and exhaustive.

$P(Y)$ - picking a yellow ball from this bag $P(notY)$ - not picking a yellow ball from this bag

What is $P(Y)$? Give your answer as a fraction.

Now we know that $P(Y)=\frac{2}{10}$ what is $P(notY)$? Give your answer as a fraction.

There is a $25\%$ chance of rain at 9am. What is the probability that it won't rain? Give your answer as a percentage.

Peter has worked out that his football team win $40\%$ of the time. What is the probability that they will lose the next match?

We have found we cannot find the probability of losing the next football match if we only know the probability of winning.

This is because winning and losing are **mutually exclusive**, but they are not **exhaustive** as it is also possible to draw.

Peter finds more data and works out that his football team win $40\%$ of the time and draw $25\%$ of the time. What is the probability that they will lose the next match?

For the probability of mutually exclusive events to add to $1$ they must be exhaustive.

In a football match there are three possible mutually exclusive events - win, lose, draw. In tossing a coin there are two - heads, tails.

Summary! Mutually exclusive events are events that cannot happen at the same time.

When rolling one die we cannot get a 5 and a 6 at the same time.

The sum of exhaustive mutually exclusive events is $1$.

We will either pick a red ball or a green ball from this bag. $P(R)+P(G)=\frac{6}{7}+\frac{1}{7}=1$.

Knowing that the sum of exhaustive mutually exclusive events is $1$ means we can work out the probability of something not happening.

$P(not6)=1-P(6)$ which is $1-P(6)=1-\frac{1}{6}=\frac{5}{6}$.