Find the Probability That an Event Will Not Happen
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 [latex]E[/latex], denoted [latex]{E}^{\prime }[/latex], is the set of outcomes in the sample space that are not in [latex]E[/latex].
the complement rule
The complement of an event [latex]E[/latex], denoted [latex]{E}^{\prime }[/latex], is the set of outcomes in the sample space that are not in [latex]E[/latex].
The probability that the complement of an event will occur is given by
To find the probability for a complement of an event, we need to use the fact that the sum of all probabilities in a probability model must be [latex]1[/latex]. This is why the rule subtracted the probability of the event from [latex]1[/latex].Suppose we are interested in the probability that a horse will lose a race. If event [latex]W[/latex] is the horse winning the race, then the complement of event [latex]W[/latex] 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 [latex]1[/latex].
The probability of the horse winning added to the probability of the horse losing must be equal to [latex]1[/latex]. Therefore, if the probability of the horse winning the race is [latex]\frac{1}{9}[/latex], the probability of the horse losing the race is simply
[latex]1-\dfrac{1}{9}=\dfrac{8}{9}[/latex]
Two six-sided number cubes are rolled.
Find the probability that the sum of the numbers rolled is less than or equal to [latex]3[/latex].
Find the probability that the sum of the numbers rolled is greater than [latex]3[/latex].
We need to count the number of ways to roll a sum of [latex]3[/latex] or less. These would include the following outcomes: [latex]1-1[/latex], [latex]1-2[/latex], and [latex]2-1[/latex].
So, there are only three ways to roll a sum of [latex]3[/latex] or less. The probability is
[latex]\dfrac{3}{36}=\dfrac{1}{12}[/latex]
Rather than listing all the possibilities, we can use the Complement Rule. Because we have already found the probability of the complement of this event, we can simply subtract that probability from 1 to find the probability that the sum of the numbers rolled is greater than [latex]3[/latex].
