Markov Chains Say the Probability of a Winning Any Point Is 2/3
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A recurring theme in the mathematics of sports Doug Ensley Shippensburg University [email protected] April is Mathematics Awareness Month! April is also… •Alcohol Awareness Month •National Oral Health Month •Stress Awareness Month •Jazz Appreciation Month •Train Safety Month Esoterica(pedia) The longest known singles tennis game was one of 80 points between Anthony Fawcett (Rhodesia) and Keith Glass (Great Britain) in the first round of the Surrey, Great Britain Championships on 26 May 1975. QUESTION: What is the probability of this happening by chance? What assumptions on a model of a tennis game can account for this freakish phenomenon? Scoring in tennis Essentially a game in tennis is won by the first player with 4 points, but that player must win by 2 points. When the score is tied 3-3, 4-4, etc., we say the score is at deuce. After a deuce score, when the server is up one point, we say the score is ad in, and when the receiver is up one point, the score is ad out. The 2005 Fawcett-Glass game had deuce 37 times. Expected Value Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Background: For a random (quantitative, discrete) variable X (e.g., number of points in a tennis game), the expected value of X is a weighted average of the possible values of X; specifically, if the possible values of X are v0, v1, v2, …, then E(X ) (vk )Pr(X vk ) k Aside: Average Value Suppose for the experiment of “choosing a random member of the Ensley family” on 04/09/2010, we define the variable X = the age of the person chosen. The following table shows the four possible values of X as well as the probability each is chosen. Value 14 17 45 46 Pr(X=Value) 0.25 0.25 0.25 0.25 Aside: Average Value Value 14 17 45 46 Pr(X=Value) 0.25 0.25 0.25 0.25 What is the average age of people in the Ensley house today? E(X ) (14) (0.25) (17) (0.25) (45) (0.25) (46) (0.25) 30.5 Tennis, anyone? Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Let X = the number of points that are played after deuce. What is the set of all possible values of X? Tennis, anyone? Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Let X = the number of points that are played after deuce. What is the set of all possible values of X? According to the definition, the expected value is the infinite series E(X ) (k)Pr(X k) k0 NOTE: The distribution of the values of X is called a geometric distribution in probability theory. Tennis, anyone? Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Solution. Let S be the set of all outcomes of this experiment. That is, S = {AA, BB, ABAA, ABBB, BAAA, BABB, ABBAAA, …} Hence, every element of S is either • AA alone, or • BB alone, or • of the form AB____ or • of the form BA____ , where the blank is filled by any element of S. Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Solution. Let S be the set of all outcomes of this experiment. That is, S = {AA, BB, ABAA, ABBB, BAAA, BABB, …} Let L be the average length of a string in S. • AA alone, • or BB alone, • or AB____, • or BA____. Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Solution. Let S be the set of all outcomes of this experiment. That is, S = {AA, BB, ABAA, ABBB, BAAA, BABB, …} Let L be the average length of a string in S. • AA alone, Probability: p∙p = p2 Length: 2 • or BB alone, • or AB____, • or BA____. Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Solution. Let S be the set of all outcomes of this experiment. That is, S = {AA, BB, ABAA, ABBB, BAAA, BABB, …} Let L be the average length of a string in S. • AA alone, Probability: p∙p = p2 Length: 2 Probability: (1 – p)2 • or BB alone, Length: 2 • or AB____, • or BA____. Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Solution. Let S be the set of all outcomes of this experiment. That is, S = {AA, BB, ABAA, ABBB, BAAA, BABB, …} Let L be the average length of a string in S. • AA alone, Probability: p∙p = p2 Length: 2 Probability: (1 – p)2 • or BB alone, Length: 2 Probability: p∙(1 – p) • or AB____, Length: 2 + L • or BA____. Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Solution. Let S be the set of all outcomes of this experiment. That is, S = {AA, BB, ABAA, ABBB, BAAA, BABB, …} Let L be the average length of a string in S. • AA alone, Probability: p∙p = p2 Length: 2 Probability: (1 – p)2 • or BB alone, Length: 2 Probability: p∙(1 – p) • or AB____, Length: 2 + L Probability: (1 – p)∙p • or BA____. Length: 2 + L Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a point with probability p? Solution. S = {AA, BB, ABAA, ABBB, BAAA, BABB, …} Elements of S (Probability)∙(Length) • AA alone p2 (2) • or BB alone, (1 – p)2 (2) • or AB____, p (1 – p) (2 + L) • or BA____. (1 – p) p (2 + L) The average length L of elements of S satisfies the equation L = p2 (2) + (1 – p)2 (2) + 2 p (1 – p) (2 + L) which has solution 2 2 L 2p2 2p 1 p2 (1 p)2 Average length of a tennis game beyond “deuce” 2 L p2 (1 p)2 NOTE: Probability theory tells us that the variance of the geometric distribution of X is given by 8p(1-p)/(p2+(1-p)2)2, which has maximum value of 8. A more general problem In tennis a deuce point is always served from the right-hand service court; an ad point is always served from the left-hand service court. Tennis broadcasts often present data on players as if there is no difference. While this might be sound at the highest levels of tennis, it is certainly not true for amateur players. We will try the previous solution method allowing for p and q to differ. Seriously? A more general problem Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a deuce point with probability p and an ad point with probability q? A more general problem Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a deuce point with probability p and an ad point with probability q? Solution. S = {AA, BB, ABAA, ABBB, BAAA, BABB, …} Every element of S is either • AA alone, or • BB alone, or • of the form AB____ , or • of the form BA____ , where the blank is filled by any element of S. Problem. What is the expected length of a tennis game which begins tied at deuce and in which player A wins a deuce point with probability p and an ad point with probability q? Solution. S = {AA, BB, ABAA, ABBB, BAAA, BABB, …} Elements of S (Probability)∙(Length) • AA alone p∙q∙(2) • or BB alone, (1 – p)∙(1 – q)∙(2) • or AB____, p∙(1 – q)∙(2 + L) • or BA____. (1 – p)∙q∙(2 + L) The average length L of elements of S satisfies the equation L = 2pq + 2(1–p)(1–q) + p(1–q)(2+L) + (1–p)q(2+L) which has solution 2 2 L 2pq p q 1 pq (1 p)(1 q) 2 f ( p,q) pq (1 p)(1 q) Examples Theorem. The expected length L of a tennis game which begins tied at deuce and in which player A wins a deuce point with probability p and an ad point with probability q is given by When q = 0.40, the maximum L is 5 with variance = 6.0. When q = 0.30, the maximum L is 6.7 with variance = 9.3. When q = 0.20, the maximum L is 10 with variance = 16.0. 2 2 When q = 0.10, the maximumL L is 20 with variance = 36.0. Even in the last, extreme case a 742pq-point p game q is1 9 standardpq (1 deviationsp)(1 q) above the mean. Alternative game scoring Some tennis matches or leagues employ "No-Ad" scoring. Each game proceeds as in regular tennis scoring, but if the score reaches deuce, then the winner of the next point, the seventh in the game, wins the game.