In basic probability, we often work with multiple events, not just one. To start this discussion, let's learn about unions and intersections. A union is indicated by the word "or". For example, what is the probability that Event A or Event B happens? That means that outcomes in either group would be desired. In the Venn Diagram below, that means anything in the colored section would be considered a desired outcome. Anything outside of the circles, but still inside the sample space would not be a desired outcome.
An intersection is indicated by the word "and". For example, what is the probability that Event A and Event B happen? This refers to outcomes that are part of both groups, but not just one group. In the Venn Diagram below, that means only the outcomes contained in the overlap of the two circles would be considered desired outcomes. Anything in the white, teal, or light purple spaces would not be desired outcomes.
Computing Probability
Example 1
A janitor has 75 keys on her keychain. Of these, 60 open classroom doors, 10 open teacher spaces, and 15 can open classrooms and teacher spaces.
Let's first find the probability that a key opens a classroom and teacher spaces. The word "and" indicates that only keys that can open both doors will be considered a success. There are 15 such keys. Basic probability says that the number of successes goes over the total number of outcomes. In this case, that's the total number of keys: 75. Therefore, the probability of picking a key that opens a classroom and a teacher space is 15/75 = .2
Example 2
Let's consider the same scenario, but this time we want to find the probability that it will open a classroom door or a teacher space. The word "or" indicates that keys that can open a classroom are a successful outcome. Keys that open a teacher space are also a successful outcome. Since these events are not mutually exclusive, that is there are keys that can open both doors, we must remember to account for the overlap of the two events. Let's break this one down into steps:
Find the probability of the first event: the key opens a classroom
There are 60 keys that open a classroom and 75 keys total. Therefore, the probability = 60 / 75 = .8
Find the probability of the second event: the key opens a teacher space
There are 10 keys that open a teacher space and 75 keys total. Therefore, the probability = 10 / 75 = .133
Find the probability of the intersection of the events: the key opens a classroom and a teacher space
This one we worked through above and found the probability to be 15 / 75 or .2
Now we're ready to plug into the formula on the probability rules page:
P(classroom) + P(teacher space) - P(intersection) = .8 + .133 - .2 = .733
