Probability of Compound Events
If we wanted to know the probability that a randomly chosen student from our class was born in either January or July, all we would need to do is add up the number of students born in January and the number of students born in July and then divide that sum by the total number of students. Does that same strategy work if we want to find the probability that a randomly chosen student is born in July or is a junior?
We could still start the same way. Let’s say the total number of students born in July is \(3\) out of a class of \(28\), and let’s say there are \(19\) juniors in that same class. The problem comes in with a student who may be a junior and born in July, because that student would be counted twice (both in the \(3\) students and in the \(19\) students). We have to take this overlap into account and subtract it off. Pretend there are \(3\) students in that class who are both junior and born in July. Essentially what we would do in this case is add the probability of July or junior and then subtract off the probability of being both a junior and born in July.
\(P\left(\text{July or junior}\right)=P\left(\text{July}\right)+P\left(\text{junior}\right)-P\left(\text{July AND junior}\right)\)
\(\large\frac{4}{28}+\frac{19}{28}-\frac{3}{28}=\frac{20}{28}=\frac{5}{7}\ \normalsize\text{or}\ 71.43%\)
These are examples of compound events which is the union or intersection of multiple events. This can be done with as many events as needed, but we will stick with just two events. You can have mutually exclusive events where there is no overlap (like in the example of being born in January or July since there is no way a person could fall into both categories) and non-mutually exclusive events where there can be overlap (like the example of being born in July or being a junior since a student could fall into both categories).
If we wanted to know the probability that a randomly chosen student from our class was born in either January or July, all we would need to do is add up the number of students born in January and the number of students born in July and then divide that sum by the total number of students. Does that same strategy work if we want to find the probability that a randomly chosen student is born in July or is a junior?
We could still start the same way. Let’s say the total number of students born in July is \(3\) out of a class of \(28\), and let’s say there are \(19\) juniors in that same class. The problem comes in with a student who may be a junior and born in July, because that student would be counted twice (both in the \(3\) students and in the \(19\) students). We have to take this overlap into account and subtract it off. Pretend there are \(3\) students in that class who are both junior and born in July. Essentially what we would do in this case is add the probability of July or junior and then subtract off the probability of being both a junior and born in July.
\(P\left(\text{July or junior}\right)=P\left(\text{July}\right)+P\left(\text{junior}\right)-P\left(\text{July AND junior}\right)\)
\(\large\frac{4}{28}+\frac{19}{28}-\frac{3}{28}=\frac{20}{28}=\frac{5}{7}\ \normalsize\text{or}\ 71.43%\)
These are examples of compound events which is the union or intersection of multiple events. This can be done with as many events as needed, but we will stick with just two events. You can have mutually exclusive events where there is no overlap (like in the example of being born in January or July since there is no way a person could fall into both categories) and non-mutually exclusive events where there can be overlap (like the example of being born in July or being a junior since a student could fall into both categories).
Example 1: One card is randomly selected from a standard deck of \(52\) playing cards. Find each probability.
a) \(P\left(8\ \text{or a black card}\right)\) b) \(P\left(\text{even number card or a queen}\right)\)
Example 2: A film company plans to release \(25\) movies in the coming year. Only \(15\) of those movies will be filmed in the USA, and \(12\) of the movies will have a female lead role. \(75\%\) of the movies with a female lead role will be filmed in the USA. Find the probability of a randomly selected movie released from this company in the coming year being filmed in the USA or having a female lead role.