A bag has r red balls and b black balls. All balls are identical except for their colours. In a trial, a ball is randoml...

Question

A bag has r red balls and b black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into the bag along with another ball of the same colour. Note that the number of balls in the bag will increase by one after each trial. A sequence of four such trials is conducted. What is the probability of drawing a red ball in the fourth trial?

A. r / (r + b)
B. (r + 3) / (r + b + 3)
C. (r + b + 3) / r
D. r / (r + b + 3)

Accepted Answer

P(red on 4th trial) = r / (r + b)   [Option A] In a Polya urn, at each step you draw a ball and return it along with a new ball of the same colour. Despite the bag growing, the probability of drawing red at any trial always equals the initial fraction r/(r+b). This is because the Polya urn model forms a martingale: the expected proportion of red balls never changes. The fourth trial is no different from the first — the answer is simply r/(r+b).