Question

An urn contains 10 balls, 3 of which are white. Three players - A, B, and C - successively draw from the urn. Each ball is replaced after it is drawn. The winner is the first one to draw a white ball. (a) Find the probability that A wins. (b) Find the probability that B wins.

Answer 1

The probability of drawing a white ball is .

p=3/10 = 0.3

a)

A wins in the following ways,

-A draws a white ball with probability

-A,B,C all do not draw a white ball and then A draws a white ball with probability

(1 - pp

-on the 7-th draw  with probability

(1 - pp

-on the 10-th draw  with probability

(1 -pºp

and so on,

The probability that A wins is the sum of all the above probabilities,

P(A) = P + (1 - pºp+ (1 - pløp+ (1 - pºp + ...... P(A) – T-110.3 p) P(A) = T-(1 – 0.3) P(A) = 0.4566

b)

B wins in the following ways,

-A does not draw a white ball and B draws a white ball with probability

(1 - pp

-A,B,C,A all do not draw a white ball and then A draws a white ball with probability

(1 - pp

-on the 8-th draw  with probability

(1 - pp

-on the 11-th draw  with probability

(1 -pop

and so on,

The probability that B wins is the sum of all the above probabilities,

P(B) = (1 - p)p+ (1 - p)p+ (1 -p) p+ (1 - p ºp+ ...... P(B) = P(1 - p) • 1 - (1 - p)3. 11- (1 - 0.3)3 P(B) = 0.3196 P(B) - 0.311 -0.3)