Bayesian Methods 9.9 Zeros on a Roulette Wheel: We now wish to make a more sophisticated estimate of the number of zeros on a roulette wheel without looking at the wheel. We will do so with Bayesian inference. Recall that when a ball on a roulette wheel falls into a non-zero slot, odd/even bets are paid; when it falls into a zero slot, they are not paid. There are 36 non-zero slots on the wheel. We assume that the number of zeros is one of {0, 1, 2, 3}. We assume that these cases have prior probability {0.1,0.2, 0.4, 0.3}. (a) Write n for the event that, in a single spin of the wheel, an odd/even bet will not be paid (equivalently, the ball lands in one of the zeros). Write z for the number of zeros in the wheel. What is P(n|z) for each of the possible values of z (i.e. each of {0, 1, 2, 3})? (b) Under what circumstances is P(z = 0|observations) NOT 0? (c) You observe 36 independent spins of the same wheel. A zero comes up in 2 of these spins. What is P(z|observations)?