Question

An unfair coin with​ Pr[H] = 0.6 is flipped. If the flip results in a​ head, a marble is selected at random from a urn containing 4 red and 6 blue marbles.​ Otherwise, a marble is selected from a different urn containing three red and five blue marbles. If the selected marble selected is​ red, what is the probability that the flip resulted in a​ head?

Answer 1

0.6154

Explanation:

Using Baye's theorem, the probability that the flip resulted in a​ head If the selected marble selected is​ red is given by;

P( H | R ) = (P( R | H ) × P( H ))/[(P( R | H ) × P( H)) + (P( R | -H ) × P( -H ))]

We can deduce that;

P( R | H ) = 4/10 = 0.4

P( R | -H ) = 3/8 = 0.375

P( -H ) = 1 - P( H )

We are given P(H) = 0.6

Thus;

P( -H ) = 1 - P( H ) = 1 - 0.6 = 0.4

Plugging in the derived values into the derived Baye's theorem, we have;

P( H | R ) = [0.4 × 0.6] /[(0.4 × 0.6) + (0.375 × 0.4)]

P( H | R ) = 0.6154