Question

An urn contains 4 white marbles and 1 black marble. Alex, Ben, and Cole take turns drawing a marble from the urn with replacement, in the order Alex, then Ben, then Cole. The first person to draw the black marble is eliminated from the game.

a) Find the probability that Ben is eliminated first.

b) After the first person is eliminated, the game continues with the two remaining players. Find the probability that Alex wins the whole game.

Answer 1

a) Probability that Ben is eliminated first is computed here as:
= Prob. that Alex takes out out a non black marble * Probability that Ben takes out a black marble + Probability that the first 4 draws are non black * Probability that the fifth draw is a black marble + ...... + Infinity

Therefore 0.3279 is the required probability here.

b) Probability that Cole is eliminated first is computed here first as:

Now probability that Alex wins the game is computed here as:

= Probability that Ben is eliminated first * Probability that next Cole is eliminated + Probability that Cole is eliminated first * Probability that Ben is eliminated next

Therefore 0.2987 is the required probability here.