Question

Two players play each other in a pool tournament of "Solids and Stripes". The first player to win two games wins the tournament. In the game of "Solids and Stripes", it is equally likely that a player will be assigned solid balls or striped balls. Assume that 1) one-half of the balls are solids and the other half are stripes, 2) the two players have the same skill: each with a 0.5 probability of winning, 3) there are no ties, and 4) the tournament is concluded once a player has won two games. In a tournament, what is the probability that a player is assigned the same ball design (i.e., solids or stripes) throughout the tournament?

Answer 1

As there are no ties in the games, there can be a maximum of three games between the two players. The results can be as follows:

X = Winners (Game 1, Game 2, Game 3)	P(X)	P(Y), Y = Same Ball Design	P(X)*P(Y)
(1,1)	0.25	0.5*0.5 (both solids) + 0.5*0.5 (both stripes) = 0.50	0.125
(1,2,1)	0.125	0.125 (solids thrice) + 0.125 (stripes thrice) = 0.25	0.03125
(1,2,2)	0.125	0.25	0.03125
(2,1,1)	0.125	0.25	0.03125
(2,1,2)	0.125	0.25	0.03125
(2,2)	0.25	0.5	0.125
		Total	0.375

P(X) is the probability of the occurrence of that particular event.

P(Y) is the probability that a player has the same ball design throughout the particular event.

For Example: X = (1, 2, 2)
Player 1 wins the first game while player 2 wins the next two games.
Probability of the occurrence = 0.5*0.5*0.5 = 0.125
Probability that a player has the same ball design throughout = 0.5*0.5*0.5 (solids throughout) + 0.5*0.5*0.5 (stripes throughout)= 0.25
Probability of occurrence of both the events = 0.25*0.125 = 0.03125

Similarly, we have calculated for all the events.

Total Probability = Sum = 0.375 = 3/8