Using Conditional Probability to Compute Probability of Intersection

The conditional probability of an event is the probability that an event A occurs given that another event B has already occurred. This type of probability is calculated by restricting the sample space that we’re working with to only the set B.

The formula for conditional probability can be rewritten using some basic algebra. Instead of the formula:

P(A | B) = P(A ∩ B) /P( B ),

we multiply both sides by P( B ) and obtain the equivalent formula:

P(A | B) x P( B) = P(A ∩ B).

We can then use this formula to find the probability that two events occur by using the conditional probability.

Use of Formula

This version of the formula is most useful when we know the conditional probability of A given B as well as the probability of the event B. If this is the case, then we can calculate the probability of the intersection of A given B by simply multiplying two other probabilities. The probability of the intersection of two events is an important number because it is the probability that both events occur.

Examples

For our first example, suppose that we know the following values for probabilities: P(A | B) = 0.8 and P( B ) = 0.5. The probability P(A ∩ B) = 0.8 x 0.5 = 0.4.

While the above example shows how the formula works, it may not be the most illuminating as to how useful the above formula is. So we will consider another example. There is a high school with 400 students, of which 120 are male and 280 are female. Of the males, 60% are currently enrolled in a mathematics course. Of the females, 80% are currently enrolled in a mathematics course. What is the probability that a randomly selected student is a female who is enrolled in a mathematics course?

Here we let F denote the event “Selected student is a female” and M the event “Selected student is enrolled in a mathematics course.” We need to determine the probability of the intersection of these two events, or P(M ∩ F).

The above formula shows us that P(M ∩ F) = P( M|F ) x P( F ). The probability that a female is selected is P( F ) = 280/400 = 70%. The conditional probability that the student selected is enrolled in a mathematics course, given that a female has been selected is P( M|F ) = 80%. We multiply these probabilities together and see that we have an 80% x 70% = 56% probability of selecting a female student who is enrolled in a mathematics course.

Test for Independence

The above formula relating conditional probability and the probability of intersection gives us an easy way to tell if we are dealing with two independent events. Since events A and B are independent if P(A | B) = P( A ), it follows from the above formula that events A and B are independent if and only if:

P( A ) x P( B ) = P(A ∩ B)

So if we know that P( A ) = 0.5, P( B ) = 0.6 and P(A ∩ B) = 0.2, without knowing anything else we can determine that these events are not independent. We know this because P( A ) x P( B ) = 0.5 x 0.6 = 0.3.  This is not the probability of the intersection of A and B.