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Nitya Nigam If you follow tennis, you've probably seen matches where games stretch into a sixth or seventh deuce, seemingly going on forever. Theoretically, if players alternated winning points from deuce, a game could last ad infinitum - the current record is 37 deuces, set in 1975. In this article, we'll find the probability that player A wins a game of tennis from deuce, given that the probability they win a point is p. This is an obvious simplification, as the probability of a player winning a point would not stay the same throughout a match, but it makes things clear in terms of the maths. For our readers who are unfamiliar with the tennis scoring system, you can find a clear and concise guide here. Firstly, let us define some probabilities. Let P_D be the probability that player A wins given that the game is at deuce. Let P_Ba be the probability that player A wins given that player B is at AD. Let P_Aa be the probability that player A wins given that player A is at AD. Now, we can use conditional probability rules to find three equations in terms of these three probabilities and our base probability p, allowing us to solve for P_D. P_D = (prob. A wins first point)(prob. A wins from AD) + (prob. B wins first point)(prob. A wins from B at AD). The probability A wins the first point is just p, so the probability B wins the first point is 1-p. Therefore: P_D = p P_Aa + (1-p) P_Ba The conditional probability equation states that P(A|B) = P(A and B) / P(B), where P(A|B) refers to the probability of event A occurring given that event B has already occurred. In our case, P_Ba = P(A wins point | B at AD) = (p x (1-p) x P_D) / (1-p) = p P_D. Similarly, P_Aa = P(A wins point | A at AD) = (p x p x P_D) / (p) Substituting these into our equation for P_D gives: When we graph this equation, we get the following shape: The shape of this graph is quite interesting. When p is 0.5, the probability of winning from deuce is also 0.5, so chances are even. But if p differs even slightly from 0.5, the chances of winning from deuce increase or decrease quite significantly. The shape of this graph demonstrates that players need to have very similar point probabilities for a game to be close. Given that so many matches in today's climate are very close, this shows just how close to each other the players are in ability, and how even a slight edge can have huge consequences.
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