A prisoner is exiled to siberia or the urals a mountainious

A prisoner is exiled to siberia or the urals (a mountainious region in russia). The chance of being sent to siberia is 70%. The prisoner knows that a guard in siberia, given randomly selcted, has a probability of .9 to be carrying a AN-94 machine gun, whereas the probability is .6 that the urals will be armed with one.

a) Upon arrival, the first guard whom the prisoner sees is armed with an AN-94. What is the probability that the prisoner has been sent to siberia?

b) Upon arrival, the first guard whom the prisioner sees is armed with an AN-94. What is the probability that the prisioner has been sent to the urals?

Solution

P(Siberia) = 0.7

P(urals) = 0.3

P(Siberia and Armed) = 0.9

P(Urals and Armed) = 0.6

a.) P(Armed) = (0.7)*(0.9) + (0.3)*(0.6) = 0.81

P(Siberia|Armed) = (0.7)*(0.9)/P(armed) = (0.7)*(0.9)/0.81 = 7/9 = 0.7778

a.) P(Armed) = (0.7)*(0.9) + (0.3)*(0.6) = 0.81

P(Urals|Armed) = (0.3)*(0.6)/P(armed) = (0.3)*(0.6)/0.81 = 2/9 = 0.2222