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o-MODELLING PROBABILITIES IN GAME OF TENNIS.
Description.
The purpose of this task is to analyze the relationship between certain probability distributions and the process of match play in tennis. We will construct a probability model first for two players of predictable abilities. We will also investigate two models of resolving ties.
Method.
Part1: Club practice.
Suppose two players have played against each other often enough to know that Adam wins about twice as many points as Ben does. The two decide to play 10 points of practice at their club.
What would be appropriate model for the distribution of X, the number of points won by Adam? Do you have any concerns about its validity? Are there limitations to its value?
Suppose a point that Adam wins is designated as A and that Ben wins as B. Using the distribution you have chosen, calculate all possible values of the random variable X. The number of points won by Adam, and draw a histogram from your chart. Document any technology that you are using to help with the calculations and /or graphing.
Find the expected value and standard deviation of this distribution. In this context, what can you say about what usually happens in these 10-point practice?
Part2. Non-extended pay games.
2. When Adam and Ben play against each other in club events, their probabilities of winning points are approximately the same as above(Adam wins about twice as many points as Ben does). In club play, the tennis rules are generally followed (win with at least four points and by at least two points in each game), but to save court time, no game is allowed to go beyond 7 points. This means that if Deuce is called (each player has 3 points), the next point determines the winner. Show that there are 70 possible ways that such a game might be played. To assist with this let Y be the number of points played. What values can Y take? For each possible value of Y find the number of possible ways that such a game could be played, and show the probability model for such a game. Be sure to define a random variable for the distribution.
Using the model you developed in 2, what is the probability that Adam wins the game? What are the odds that he wins?
Generalise this to find the probability that Player C wins in terms of c and d, where c represents the probability that Player C wins a point and d represents the probability that Player D wins a point.
Part 3 : Extended play games.
When Adam and Ben play against each other in tournaments outside the club, their point-winning probabilities remain the same (2/3 and 1/3, respectively), but the rules now require that players win by 2 points and therefore, that games may in theory be infinitely long.
Show that although Adams point odds against Ben are 2:1, his game odds are almost 6:1. Be sure to consider separately the cases of non-deuce and deuce games.
Suppose that, more generally, Player Cs probability of winning a point is c and player Ds probability of winning a point is d. Write formulas in terms of c and d for their probabilities: probability that player C wins without deuce being called, probability that deuce is called, and probability that Player C wins given that deuce is called. Using these formulas to aid calculation, find the odds that Player C wins for c=0.5, 0.55, 0.6, 0.7, 0.9 and any other values you would like to test. A spreadsheet is encouraged. C4 + D
What expression represents the odds in such situations? What happens when the winning probabilities are close together or when one player is almost certain to win each point?
Evaluate the usefulness and limitations of such probability models.
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