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 5 red, 2 white, and 6 blue marbles. If the flip results in a tail then a marble is selected from an urn containing 9 red and 3 white marbles. If the marble selected is white, then what is the probability that a flip resulted in a head?

Answer 1

Let's Use three events.

 

H: Heads tossed

T: Tails tossed

W: White marble chosen

 

 

Note that

 

Pr(W|H) = 2 / (5+2+6) = 2/13

 

Pr(W|T) = 3 / (9 + 3 + 0) = 3/12 = 1/4

 

 

We can now use Bayes' rule.

 

Pr(H|W) = Pr(W|H) Pr(H) / [ Pr(W|H) Pr(H) + Pr(W|T) Pr(T)]

 

= (2/13)(2/3) / [(2/13)(2/3) + (1/4)(1/3)]

 

= (4/39) / (4/39 + 1/12)

 

= (4/39) / (16/156 + 13/156)

 

= (4/39) / (29/156)

 

= (4·156) / (29·39)

 

= 16/29