Here, you will learn formula for conditional probability and properties of conditional probability with examples.
Let’s begin –
Formula for Conditional Probability
Let A and B be two events associated with a random experiment. Then, the probability of occurrence of event A under the condition that B has already occured and P(B) \(\ne\) 0, is called the conditional probability and it is denoted by P(A/B). Thus, we have
P(A/B) = Probability of occurrence of A given that B has already occurred
P(A/B) = \({P(A\cap B)}\over P(B)\) = which is called Conditional Probability of A given B.
Similarly, P(B/A) when P(A) \(\ne\) 0 is defined as the probability of occurrence of event B when A has already occurred.
P(B/A) = \({P(A\cap B)}\over P(A)\) = which is called Conditional Probability of B given A.
Example : Let there be a bag containing 5 white and 4 red balls. Two balls are drawn from the bag one after the other without replacement.
Solution : Consider the following events :
A = Drawing a white ball in the first draw,
B = Drawing a red ball in the second draw
Now,
P(B/A) = Probability of drawing a red ball in second draw given that a white ball has already been drawn in the first draw
\(\implies\) P(B/A) = Probability of drawing a red ball from a bag containing 4 white and red balls
\(\implies\) P(B/A) = \(4\over 8\) = \(1\over 2\)
For this random experiment P(A/B) is not meaningful because A cannot occur after the occurence of event B.
Properties of Conditional Probability
(i) Let A and B be two events associated with sample space S, then 0 \(\le\) P(A/B) \(\le\) 1.
(ii) If A is an event associated with the sample space S of a random experiment, then P(S/A) = P(A/A) = 1
(iii) Let A and B be two events associated with a random experiment and S be the sample space, if C is an evnt such that P(C) \(\ne\) 0, then
P(\((A\cup B)/C\)) = P(A/C) + P(B/C) – P(\((A\cap B)/C\))
In Particular, if A and B are mutually exclusive events, then
P(\((A\cup B)/C\)) = P(A/C) + P(B/C)
(iv) If A and B are two events associated with a random experiment, the P(A’/B) = 1 – P(A/B)