Question

Suppose the probability of a server winning any given point in a tennis match is a constant \(p,\) with \(0 \leq p \leq 1\) Then the probability of the server winning a game when serving from deuce is $$f(p)=\frac{p^{2}}{1-2 p(1-p)}$$ a. Evaluate \(f(0.75)\) and intepret the result. b. Evaluate \(f(0.25)\) and intepret the result.

Short answer

Answer: If the server has a probability of 0.75 of winning any given point, the probability of winning the game when serving from deuce is 90%. On the other hand, if the server has a probability of 0.25 of winning any given point, the probability of winning the game when serving from deuce is 10%.

Step-by-step solution

Evaluate f(0.75)

To evaluate f(0.75), plug p = 0.75 into the given formula: $$f(0.75) = \frac{(0.75)^2}{1 - 2(0.75)(1 - 0.75)}$$

Calculate Numerator

First, calculate the numerator: $$(0.75)^2 = 0.5625$$

Calculate Denominator

Next, calculate the denominator: $$1 - 2(0.75)(1 - 0.75) = 1- 1.5(0.25) = 1 - 0.375 = 0.625$$

Divide Numerator by Denominator

Divide the numerator by the denominator: $$f(0.75) = \frac{0.5625}{0.625} = 0.9$$

Interpret the Result

The result f(0.75) = 0.9 means that if the server has a probability of 0.75 of winning any given point, the probability of winning the game when serving from deuce is 90%.

Evaluate f(0.25)

To evaluate f(0.25), plug p = 0.25 into the given formula: $$f(0.25) = \frac{(0.25)^2}{1 - 2(0.25)(1 - 0.25)}$$

Calculate Numerator

First, calculate the numerator: $$(0.25)^2 = 0.0625$$

Calculate Denominator

Next, calculate the denominator: $$1 - 2(0.25)(1 - 0.25) = 1 - 0.5(0.75) = 1 - 0.375 = 0.625$$

Divide Numerator by Denominator

Divide the numerator by the denominator: $$f(0.25) = \frac{0.0625}{0.625} = 0.1$$

Interpret the Result

The result f(0.25) = 0.1 means that if the server has a probability of 0.25 of winning any given point, the probability of winning the game when serving from deuce is 10%.

Key concepts

Tennis

In tennis, players compete to win games, sets, and ultimately, the match. Each game is composed of points, which follow the unique scoring system of love (0), 15, 30, 40, and then game. When the score reaches "deuce" (40-40), it means that two consecutive points are needed by a player to win the game. Thus, the concept of probability becomes crucial in determining the outcomes of competitive circumstances.

For servers in particular, winning a game from deuce can be a daunting task because it requires consistently performing well under pressure. Here, the probability of winning becomes a key metric for evaluating performance and planning strategy. By understanding the likelihood of such occurrences, players and coaches can better understand the potential outcomes and adjust their tactics accordingly.

Mathematical Function

A mathematical function is a special relationship where each input value has a unique output. In the context of our tennis exercise, the function provided is used to calculate the probability of winning a game from "deuce" based on the probability of winning a single point, denoted as \( p \). The function is conveyed in the given formula:

\[f(p)=\frac{p^{2}}{1-2p(1-p)}\]

This formula has two main components:

• The numerator \( p^2 \), representing the probability of winning two consecutive points.
• The denominator \( 1 - 2p(1-p) \), which adjusts for the chances of the opponent winning a point and redistributing the sequence.

To evaluate this function, you replace \( p \) with any given probability value within its range (0 to 1), and follow the arithmetic steps to calculate the output. This function helps predict the overall likelihood of success for a player serving at deuce.

Interpretation of Results

Interpreting the results of the function involves understanding what the output number signifies in a real-world context. In our scenario, using the formula with \( p = 0.75 \) tells us that the server has a 90% probability of winning the game from deuce. This translates to a high likelihood of success if the server is winning 75% of their points overall.

In contrast, evaluating the function with \( p = 0.25 \) results in a 10% probability of winning. It shows much less confidence in the server's ability to close the game from deuce when they win just 25% of their points.

Understanding these interpretations allows players to assess their strengths and weaknesses. A high percentage suggests a strong serve performance, while a low percentage indicates areas needing improvement. It highlights how probability measurements can aid in strategic planning, preparation, and mental conditioning in sports like tennis.