# How Is the Number of Tennis Sets Distributed in a Match?

• Benny
Then, use the normal distribution formula to find Pr(T>=70). This means you need to find the probability that the total number of sets played in the 16 matches is equal to or greater than 70. This can be calculated by finding the probability of getting 70 sets, 71 sets, 72 sets, and so on, up to 80 sets (since the total number of sets cannot exceed 80). Then, add these probabilities together to get the final answer. In summary, you need to use a normal distribution with the mean and standard deviation calculated in part (a) to find the probability that the total number of sets played in 16 matches is equal to or greater than 70.
Benny
Can someone help me work out the following question?

Q. Two tennis professionals, A and B, are scheduled to play a 'best of five sets' match, for which the winner is the first player to win three sets in a total that cannot exceed five sets. The probability that A wins anyone set is 0.6, independent of the outcome of any other set. Let X denote the number of sets played in the match. Then the distribution of X is given by:

Code:
    x      |    3          4          5
-----------------------------------
Pr(X=x) |  0.280    0.374     0.346

(a) Show that the mean and standard deviation of X are 4.066 and 0.788 (correct to three decimal places), respectively.

(b) Over a year, the two players play each other in 16, best of five sets, matches. Find the mean and standard deviation of T, the total number of sets they play in the 16 matches and, using a normal approximation with continuity correction, find Pr(T>=70).

The first one is just plugging values into formulas but I don't know how to start the second part. I think it has something to do with the binomial distribution. There is a fixed number of matches (16) but I don't know how to relate the information in part (a) to part (b).

Can someone help me out? Thanks.

I think (b) is asking you to assume a normal distribution with the mean and the standard dev. you calculated in (a).

Sure, I can help you work out the second part of the question. In order to find the mean and standard deviation of T, we need to first understand the distribution of T. Since each match can have a maximum of 5 sets, the total number of sets played in 16 matches can range from 0 to 80. This means that T follows a binomial distribution with n=16 and p=0.346 (from the distribution of X given in the question).

To find the mean of T, we use the formula μ = np, where μ is the mean, n is the number of trials and p is the probability of success. Substituting the values, we get μ = 16*0.346 = 5.536 sets.

To find the standard deviation of T, we use the formula σ = √(np(1-p)). Substituting the values, we get σ = √(16*0.346*0.654) = 2.159 sets.

To find Pr(T>=70), we can use a normal approximation with continuity correction. This means that we can approximate the binomial distribution of T with a normal distribution with the same mean and standard deviation.

Using the z-score formula, we can find the z-score for T=70 as z = (70-5.536)/2.159 = 31.56.

Now, using a z-table, we can find the probability of T being greater than or equal to 70 as P(Z>=31.56) = 1- P(Z<=31.56) = 1- 0.999999 = 0.000001.

Therefore, the probability of T being greater than or equal to 70 is extremely low, indicating that it is highly unlikely for the players to play 70 or more sets in 16 matches.

I hope this helps you understand the problem better. If you have any further questions, please feel free to ask.

## 1. What is the tennis probability problem?

The tennis probability problem is a mathematical problem that involves calculating the probability of winning a game or match in a game of tennis. It takes into account various factors such as player abilities, chance, and game rules.

## 2. What are the variables that affect the tennis probability problem?

The variables that affect the tennis probability problem include the skill level of the players, the type of court surface, the weather conditions, and the scoring system used.

## 3. How is the tennis probability problem solved?

The tennis probability problem is solved using mathematical equations and statistical analysis. The specific method used may vary depending on the specific scenario and variables involved.

## 4. How does the tennis probability problem impact the outcome of a game or match?

The tennis probability problem helps to predict the likelihood of a player winning a game or match. It can be used by coaches and players to strategize and make informed decisions during a game.

## 5. What are some real-life applications of the tennis probability problem?

The tennis probability problem can be applied in various real-life situations, such as sports betting, player performance analysis, and game strategy development. It can also be used in other sports with similar scoring systems, such as badminton or table tennis.

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