An unfair coin with Pr[H]=2/3 is flipped.

Question

An unfair coin with Pr[H] = 2/3 is flipped. If the flip results in a head, then a marble is selected from an urn containing 9 red, 7 white, and 5 blue marbles. If the flip results in a tail then a marble is selected from an urn containing 3 red and 9 white marbles. If the marble selected is white, then what is the probability that a flip resulted in a head?

Answer 1

The chance of getting a white marble from urn#1 is 7/21 = 1/3

The chance of getting a white marble from urn#2 is 9/12 = 3/4

 

Therefore, we know that (1/3) of all heads tossed will result in a white marble

We also know that (3/4) of all tails tossed will result in a white marble

 

Therefore, we can figure out the probability of heads being tossed (given the marble is white) by dividing the chance to get white from heads by the chance to get white from total:

 

p(heads) = (1/3) / (1/3 + 3/4)

= 1/3 / (4/12 + 9/12)

= 1/3 / (13/12)

= 1/3 * 12/13

= .30769...

 

Therefore, the probability is about 31%