The rules of probability

In probability, we have some rules that become very useful when visualization gets too cumbersome. These rules help us calculate compound probabilities with ease.

The addition rule

The addition rule is used to calculate the probability of either or events. To calculate P(A B) = P(A or B), we use the following formula:

P(A B) = P(A) + P(B) − P(A ∩ B)

The first part of the formula (P(A) + P(B)) makes complete sense. To get the union of the two events, we have to add together the area of the circles in the universe. But why the subtraction of P(A and B)? This is because when we add the two circles, we are adding the area of intersection twice, as shown in the following diagram:

See how both the red circles include the intersection of A and B? So, when we add them, we need to subtract just one of them to account for this, leaving us with our formula.

Recall that we wanted the number of people who either had cancer or had a positive test result? If A is the event that someone...