An unfair coin with Pr[H] = 0.2 is flipped. If the flip results in a head, a marble is selected at random from a urn containin
g six red and four 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?
Normally when dealing with coins the probability of getting heads or tails is 0.5 each. However in this case since its an unfair coin, the probability of getting heads is 0.2. H - head T - tails R - red marble pr (H) = 0.2 urn 6 red and 4 blue pr (T) = 0.8 urn 3 red and 5 blue
when heads is obtained red - 6/10 -0.6 blue - 4/10 - 0.4 therefore when multiplying with 0.2 probability of getting heads pr (R ∩ H) red - 0.6*0.2 = 0.12
when tails is obtained red - 3/8 - 0.375 blue - 5/8 - 0.625 when multiplying with 0.8 probability of getting tails pr (R ∩ T) red - 0.375 * 0.8 = 0.3
using bayes rule the answer can be found out, the following equation is used; pr (H | R) = pr (R ∩ H) / {pr (R ∩ H) + pr (R ∩ T)} = 0.12 / (0.12 + 0.3) = 0.12 / 0.42 = 0.286 the final answer is 0.286
