# 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)