Red balls pay, the yellow one ruins
An urn holds 75 blue balls, 25 red balls and 1 yellow ball. You draw balls one at a time without replacement. Each red ball earns you 1 dollar, a blue ball earns nothing, and drawing the yellow ball wipes out everything you have won and ends the game. You may stop before any draw and keep your winnings.
Find your expected winnings under the best strategy.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Blue balls change nothing. Only the order of the red balls and the yellow ball matters.
With dollars won and red balls left, compare stopping with drawing until the next red or yellow ball.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Ignore the blue balls: they neither pay nor hurt. Among the red balls and the yellow one, each draw is a bet: win 1 dollar or lose everything. Stop when the bet turns unfavourable, then compute the chance of reaching that point.
Step 1: the one-step comparison. With won, red balls and the yellow one remain among the balls that matter. The next of them is red with probability . Drawing once more is worth on average which beats exactly when , that is , or .
Step 2: the rule. Keep drawing until you have 13 dollars, then stop. Once , every further draw is unfavourable and stays so, so the one-step rule is optimal.
Step 3: its value. The yellow ball is equally likely to be in any of the 26 positions among the red and yellow balls. You collect 13 dollars when it is in position 14 or later: probability . So