Conditional Probability

Definition : If A and B are two events of same sample space of a random experiments, then the probability of occurrence of event B when event A has already occurred. It means event will occurr iff the event A has already occurred.

It is denoted as P(

B/A

) =

P(A∩B)/P(A)

, P(A) ≠ 0

Similarly, P(

A/B

) is used to denote the probability of occurrence of A when B occurs .

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Illustration : An urn contains 6 red and 9 green balls . Two balls are drawn from the urn one after another without replacement . Find the probability of drawing a red ball when a green ball has been drawn from the urn .
Let A = drawing a red ball in the second draw and B = drawing a green ball in the first draw

So P(

A/B

) =

6/14

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Properties of Conditional Probability

(1) Let A and B be events of a sample space S of an random experiment, then

P(

S/A

) = P(

A/A

) = 1

(2) P(

A'/B

) + P(

A/B

) = 1

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Independent Events :

Events are known as independent if the occurrence of one has no impact on others . Suppose an urn contain A red balls an B green balls and two balls are to be drawn one after the other . If the ball drawn in the first attempt is not replaced in the bag , then two events of drawing the ball are dependent because first draw of the ball has impact on the second ball drawing . If the ball drawn in the first attempt is replaced back in the bag, then two events are independent because first drawn of a ball has no impact on the second draw .