Probability Notes: Total Probability
My studies continue in Peter Olofsson's excellent book "Probabilities: The Little Numbers That Rule Our Lives". Here is the general formula for Total Probability, for two events A and B:
P(B) = P(B given A) × P(A) + P(B given (not A)) × P(not A)
This formula allows us to determine the probability an event occurs based on our knowledge of its dependence on another event occuring, and the probability of that other event occurring and not occuring. Interesting and entertaining examples are given in the book (pages 28 and following) but it is not my desire to start copying out here large portions of his book. But it is not too difficult to come up with another fictional example:
Suppose that you did a survey and found that 9 out of 10 college professors who teach history owned a collection of books on Roman history. Also, from another survey, you determined that, among college professors that do not teach history, only 1 out of 20 owned such a collection. Then, from statistics, you determed that 1 out of 8 college professors taught history. What is the probability that any randomly chosen professor owns a collection of books on Roman history?
9/10 × 1/8 + 1/20 × 7/8 = 5/32 ≅ 0.156
Or about 15.6 %, which is about one professor out of every 6½ professors that you (randomly) meet.
A few notes on the Total Probability formula:
(1) If the event B is independent from the event A (not affected by it) then the formula reduces to the somewhat less useful P(B) = P(B).
(2) If our related events are mutually exclusive, and cover all cases, then we can extend the formula to more than two cases. For example:
P(D) = P(D given A) × P(A) + P(D given B) × P(B) + P(D given C) × P(C)