13.7 Probability  (Page 3/18)

Computing the probability of mutually exclusive events

Suppose the spinner in [link] is spun again, but this time we are interested in the probability of spinning an orange or a $d.$ There are no sectors that are both orange and contain a $d,$ so these two events have no outcomes in common. Events are said to be mutually exclusive events    when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is

Notice that with mutually exclusive events, the intersection of $E$ and $F$ is the empty set. The probability of spinning an orange is $\frac{3}{6}=\frac{1}{2}$ and the probability of spinning a $d$ is $\frac{1}{6}.$ We can find the probability of spinning an orange or a $d$ simply by adding the two probabilities.

The probability of spinning an orange or a $d$ is $\frac{2}{3}.$

Using the complement rule to compute probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event $E,$ denoted $E^{\prime },$ is the set of outcomes in the sample space that are not in $E.$ For example, suppose we are interested in the probability that a horse will lose a race. If event $W$ is the horse winning the race, then the complement of event $W$ is the horse losing the race.

To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.

The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is $\frac{1}{9},$ the probability of the horse losing the race is simply