If two events a and b are independent and you know that p(a) = 0.85, what is the value of p(a | b)?

Solution:

p(a | b) is a fundamental relation of conditional probability relating two dependent events. Mathematically it is defined as follows:

p(a | b) = p(a and b)/p(b) --- (1)

p(a and b) = joint probability of events a and b

p(b) = marginal probability of event b

p(a | b) = probability of an event occurring when event “b” has already taken place

Now since a and b are independent events the relation gets modified as

p(a | b) = p(a) × p(b)/p(b) = p(a)

Therefore,

p(a | b) = p(a) = 0.85

p(a | b) means the probability of event “a” occurring once the event “b” has taken place. In other words, it is an expression of conditional probability. The expression necessarily assumes that both event “a” and event “b” are dependent. However, the problem statement clearly states that the two events are independent. Thus, the numerator in equation (1) can be written as p(a) × p(b). The p(b) in the numerator gets canceled by the p(b) in the denominator and p(a | b) simply gets reduced to p(a) which is the marginal property of “a” and is equal to 0.85 which is the answer to the question.

If two events a and b are independent and you know that p(a) = 0.85, what is the value of p(a | b)?

Summary:

If events a and b are independent then the value of p(a | b) = 0.85 which is the marginal probability of event a.