Two events A and B are said to be dependent when B can occur only when A is known to have occurred (or vice versa). The probability attached to such an event is called the conditional probability and is denoted byP(A/B) or, in other words probability of A given that B has occurred. If two events A and B are dependent, then the conditional probability of B given A is: P(B/A) = P(AB)/P(A) Proof: suppose a1 is the number of cases for the simultaneous happening of A and B out of a1 + a2 cases in which A can happen with or without happening of B. ∴ P(B/A) = a1/(a1 + a2) = (a1/n)/(a1 + a2)/n = P(AB)/P(A) Similarly it can be shown that P(A/B) = P[(AB)/P(B)] The general rule of multiplication in its modified form in terms of conditional probability becomes: P(A and B) = P(B) × P(A/B) Or, P(A and B) = P(A) × P(B/A) For three events A, B and C we have P(ABC) = P(A) × P(B/A) × P(C/AB) i.e. the probability of occurrence of A, B and C is equal to the probability of A, times of the probability of B given that A has occurred, times the probability of C given that both A and B have occurred. Illustration: a bag contains 5 white and 3 black balls. Two balls are drawn at random one after the other without replacement. Find the probability that both balls drawn are black. Solution: probability of drawing a black ball in the first attempt is P(A) = 3/(5 + 3) = 3/8 Probability of drawing the second black ball given that the first ball drawn is black P(B/A) = 2/(5 + 2) = 2/7 ∴ The probability that both balls drawn are black is given by P(AB) = P(A) × P(B/A) = 3/8 × 2/7 = 3/28

Posted Date: 2/14/2012 1:39:25 PM | Location : United States

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