Conditional Probability - Practice Questions & MCQ

Conditional Probability

Conditional probability is a measure of the probability of an event given that another event has already occurred. If A and B are two events associated with the same sample space of a random experiment, the conditional probability of the event A given that B has already occurred is written as  P(A|B), P(A/B) or .

The formula to calculate P(A|B) is

For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = face is 2 or 3 and B = face is even number (2, 4, 6).

Here, P(A|B) means that the probability of occurrence of face 2 or 3 when an even number has occurred which means that one of 2, 4 and 6 has occurred. 

To calculate P(A|B), we count the number of outcomes 2 or 3 in the modified sample space B = {2, 4, 6}: meaning the common part in A and B. Then we divide that by the number of outcomes in B (rather than S).

Properties of Conditional Probability

Let A and B are events of a sample space S of an experiment, then

Property 1  P(S|A) = P(A|A) = 1

Proof:

Property 2  If A and B are any two events of a sample space S and C is an event of S such that P(C) ≠ 0, then

                    P((A ∪ B)|C) = P(A|C) + P(B|C) – P((A ∩ B)|C)

In particular, if A and B are disjoint events, then

                    P((A∪B)|C) = P(A|C) + P(B|C)

Proof:

Property 3 P(A’|B) =1 - P(A|B) , if P(B) ≠ 0

Proof:

                   From Property 1, we know that P(S|B) = 1

⇒                P((A ∪ A’)|B) = 1                               (as A ∪ A’ = S )     

⇒                P(A|B) + P(A’|B) = 1                          (as A and A’ are disjoint event)

⇒                P(A’|B) = 1 - P(A|B)