# Chess Players Truel

Players A, B, and C play a series of chess games. Assume that A is the strongest player and C is the weakest one. Assume that there is no tie for each game. The winner of each game will play with the 3^{rd} player. The player who first gets 2 wins is the winner of the series. The player B determines who will play the 1^{st} game. Find the best choice for B. In general: if probability of A to win B is

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Copyright © 1996-2018 Alexander Bogomolny

Players A, B, and C play a series of chess games. Assume that A is the strongest player and C is the weakest one. Assume that there is no tie for each game. The winner of each game will play with the 3^{rd} player. The player who first gets 2 wins is the winner of the series. The player B determines who will play the 1^{st} game. Find the best choice for B. In general: if probability of A to win B is

The simplest way to approach the problem is by constructing an *event tree*. We are going to have three of them. The trees grow downwards starting with one of the pairs AB, BC or AC. Each node is a game, except for the terminal ones shown in squares. These are the winners of the tournament who collected two wins first. For the first two trees we'll find the probability of B being the winner. These will be denoted P_{AB} and P_{BC}. We assume the outcomes of all games are independent.

P_{AB} = p·(1 - r)·q·(1 - p) + (1 - p)·q + (1 - p)·(1 - q)·r·(1 - p).

P_{BC} = q·(1 - p) + q·p·(1 - r)·q + (1 - q)·r·(1 - p)·q.

The task is to compare the two expressions P_{AB} and P_{BC} assuming _{AB} < P_{BC}

What if applet does not run? |

So here are the two expressions to compare:

- P
_{AB}= p·(1 - r)·q·(1 - p) + (1 - p)·q + (1 - p)·(1 - q)·r·(1 - p). - P
_{BC}= q·(1 - p) + q·p·(1 - r)·q + (1 - q)·r·(1 - p)·q.

In the difference P_{BC} - P_{AB}, the common term (1 - p)·q cancels out leading to

P_{BC} - P_{AB}= [p·q·(1 - r) + (1 - p)·(1 - q)·r]·(p + q - 1)

which, since p + q > .5 + .5 = 1, is positive. Note that this is true even without the condition

- What Is Probability?
- Intuitive Probability
- Probability Problems
- Sample Spaces and Random Variables
- Probabilities
- Conditional Probability
- Dependent and Independent Events
- Algebra of Random Variables
- Expectation
- Probability Generating Functions
- Probability of Two Integers Being Coprime
- Random Walks
- Probabilistic Method
- Probability Paradoxes
- Symmetry Principle in Probability
- Non-transitive Dice

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