Formula for Conditional Probability

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)

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