Suppose wc need lo probabilisticalJy simulate lhe outcome of

Suppose wc need lo probabilisticalJy simulate lhe outcome of a coin Hip, i.e. thai il yields a \"Heads\" 01 \"Tails\" with equal probability. Also suppose thai the only device we have at our disposal is a 6-sided die whose fairness is dubious. You can assume that each face i of this die has an unknown probability ph 6 of occurring, where Hf=l p,- = 1. With this die. we want to simulate the outcome of a fair coin flip using the following procedure. Roll the die. Roll the die again. If the results of the two die rolls are both odd numbers or they are both even numbers, then go back to Step I, Otherwise, if the result of the last roll is an even number, take that to be \"Heads\", or else, if it\'s an odd number, take that to be \"Tails\". Answer the following questions regarding this procedure. Show that for a given number of total die rolls, the result of this procedure is equally likely to be heads or tails. Show that the probability of this procedure being repeated indefinitely is zero. i.e. that the probability of getting tails or heads each converges to V, as the number of repeated die rolls go to infinity. What is the expected number of die rolls (as a function of the p_1 ,6) until the procedure yields a result? Also show that this expected value is minimum when the die lias equal probability of returning an even and an odd number (in particular, when it is a fair die). What is the variance of the number of die rolls until the procedure yields a result?

Solution

a)

x= 1 1/6

x=2 1/6

x=3 1/6

x=4 1/6

x=5 1/6

x=6 1/6