There are two bowls of chips which are red white or blue The

There are two bowls of chips which are red, white, or blue. The first bowl B1 contains 6 of each of the chips. The second bowl B2 contains 9 red chips, 6 white chips, and 3 blue chips. A fair 4 sided die is rolled. If the outcome is [1], a chip is selected randomly from bowl B1. If the outcome belongs to [2, 3, 4], a chip is selected randomly from Bowl B2. Find (a) P(red chip), the probability of selecting a red chip, (b) P(B1|red chip), the probability that the chip was selected from bowl B1 given that the chip was red.

Solution

a) Probability of selecting red chip

By Bayes theorem = P(red from B1) + P(red from B2)

Possible ways of selecting red chip are if dice outcome is 1 and selecting 1 redchip of 6 redchips in bowl 1 with total (6+6+6) chips or if dice outcomw is 2,3,4 and selecting one red chip from 9 redchips in bowl2 with (9+6+3) chips

=((1/4)*(6/(6+6+6))) + ((3/4)*(9/(9+6+3))) = (1/12) + (3/8) = 11/24 = 0.4583

b) from the above problem P(B1/red) = P(red from B1)/P(red) = (1/12)/((1/12)+(3/8)) = 2/11 = 0.181