Finite Help: Probability

Question

Involving the selection of a colored ball from one of three bowls. Assume that you randomly pick one of the bowls: X, Y, or Z. You then randomly draw one ball out of the selected bowl and note its color. The bowls contain the following colored balls: Bowl X: 1 red, 2 white, and 2 blue. Bowl Y: 1 red, 2 white, and 4 blue. Bowl Z: 2 red, 2 white, and 2 blue. (1) If the ball is blue, what is the probability that it was drawn from bowl X? (2) If the ball is either blue or white, what is the probability that it was drawn from bowl Y:

Answer 1

(1) If the ball is blue, what is the probability it was drawn from bowl X?

Prob. Overall of drawing a blue ball = sum of # of blue balls in each bowl/ sum of all balls in all 3 bowls = (2 + 4 + 2)/(1 +2+2+1+2+4+2+2+2)

= 8/18

= 4/9

Prob. That the blue ball came from bowl X = # of blue balls in bowl X/# of blue balls in total

2/8 = 1/4

Multiply the 2 fractions together:

(4/9)×(1/4) = 4/36 = 1/9

(2) probability the ball is blue OR white = total # of blue and white balls in all bowls/ total # of balls of all colors

14/18 = 7/9

Probability that it's drawn from bowl Y = # of blue & white balls in bowl Y/ total # of blue & white balls

6/14 = 3/7

(7/9)×(3/7) = 21/63 = 1/3