Question

Two individuals, \(A\) and \(B\), are finalists for a chess championship. They will play a sequence of games, each of which can result in a win for \(\mathrm{A}\), a win for \(\mathrm{B}\), or a draw. Suppose that the outcomes of successive games are independent, with \(P(\) A wins game \()=.3, P(\) B wins game \()=.2\), and \(P(\) draw \()=.5 .\) Each time a player wins a game, he earns 1 point and his opponent earns no points. The first player to win 5 points wins the championship. For the sake of simplicity, assume that the championship will end in a draw if both players obtain 5 points at the same time. a. What is the probability that A wins the championship in just five games? b. What is the probability that it takes just five games to obtain a champion? c. If a draw earns a half-point for each player, describe how you would perform a simulation to estimate \(P(\) A wins the championship). d. If neither player earns any points from a draw, would the simulation in Part (c) take longer to perform? Explain your reasoning.

Short answer

a. The probability that A wins in just five games is .00243. b. The probability that it takes just five games to obtain a champion is .00275. c. For the simulation, we randomly generate results for each game following the probability distributions given. We add a half-point for each player for each draw and continue until one player gets 5 points. d. Yes, likely the simulations would take longer if no points are awarded for a draw since more games would be needed to reach 5 points.

Step-by-step solution

Probability calculation for part a

Based on the probability theory, the probability P(A wins a game) = .3 and winning five games in a row equals \(.3^5\) = .00243.

Probability calculation for part b

For any player to win in five games, they must win all five games. So, the calculations will be done for each player and then added together because these are independent events. \((P(A)\) wins 5 in a row + \(P(B)\) wins 5 in a row = \(.3^5 + .2^5\) = .00243 + .00032 = .00275.

Description of Simulation for part c

To simulate the championship, we would randomly generate results for each game following the probability distributions given. For every drawn game, each player would get a half point. The process would be continued until A or B reaches or exceeds 5 points. The probability that A wins the championship would be estimated as the ratio of the number of simulations in which A reaches 5 points before B to the total number of simulations.

Analyzing the effect on Simulation for part d

If draws no longer provide any points, the total number of games to reach 5 points for any player would potentially increase as fewer points would be earned per game. Therefore, the simulation would take longer to perform.