Conditional Probability

T. Dhasaratharaman*

Statistician, Kauvery Hospitals, India

*Correspondence: Tel.: +91 90037 84310; email: [email protected]

It is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read as “the probability of A, given B”.

The probability that event B occurs, given that event A has already occurred is P(B|A) = P(A and B)/P(A).

This formula comes from the general multiplication principle and a little bit of algebra.

Since we are given that event A has occurred, we have a reduced sample space. Instead of the entire sample space S, we now have a sample space of A since we know A has occurred. If you then divided the numerator and denominator of the right-hand side by the number in the sample space S, then you have the probability of A and B divided by the probability of A.

Example 1

The question, “Do you smoke?” was asked of 100 people. Results are shown in the table.

	Yes	No	Total
Male	19	41	60
Female	12	28	40
Total	31	69	100

What is the probability of a randomly selected individual being a male who smokes? This is just a joint probability.

The number of “Male and Smoke” divided by the total = 19/100 = 0.19

What is the probability of a randomly selected individual being a male? This is the total for male divided by the total = 60/100 = 0.60. Since no mention is made of smoking or not smoking, it includes all the cases.

What is the probability of a randomly selected individual smoking? Again, since no mention is made of gender, this is a marginal probability, the total who smoke divided by the total = 31/100 = 0.31.

What is the probability of a randomly selected male smoking? This time, you’re told that you have a male – think of stratified sampling. What is the probability that the male smokes? Well, 19 males smoke out of 60 males, so 19/60= 0.31666.

What is the probability that a randomly selected smoker is male? This time, you’re told that you have a smoker and asked to find the probability that the smoker is also male. There are 19 male smokers out of 31 total smokers, so 19/60= 0.6129.

Mr. T. Dhasaratharaman

Statistician

Conditional Probability