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

Bayes' Theorem:

 

P( H | W ) = P( W | H )P( H )/[P( W | H )P( H ) + P( W | -H )P( -H )]

 

where H is "heads flipped", -H is "not heads flipped" (i.e. "tails flipped"), and W is "white drawn".

P( W | H ) = 7/21 = 1/3

P( W | -H ) = 9/12 = 3/4

P( -H ) = 1 - P( H ) = 1/3

 

By plugging into the formula above,

P( H | W ) = (1/3 * 2/3) / [1/3 * 2/3 + 3/4 * 1/3]

= (2/9) / (17/36)

= 72/153

= 8/17.