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 interpret the result. b. Evaluate \(f(0.25)\) and interpret the result. (Source: The College Mathematics Journal 38, 1, Jan 2007).

Short answer

Answer: If the server has a 75% chance of winning any given point, the probability of winning a game from deuce is 90%. If the server has a 25% chance of winning any given point, the probability of winning a game from deuce is 10%.

Step-by-step solution

In this step, we will substitute \(p=0.75\) into the function \(f(p)=\frac{p^{2}}{1-2 p(1-p)}\) to get the probability of the server winning a game when serving from deuce. #Step 2: Calculate the probability of winning from deuce#

After substituting \(p=0.75\) into the function \(f(p)\), we can calculate the probability: $$f(0.75)=\frac{(0.75)^2}{1-2(0.75)(1-0.75)}$$ #Step 3: Simplify the expression#

Now, we will simplify the expression to get the final probability value: $$f(0.75)=\frac{0.5625}{1-0.375} = \frac{0.5625}{0.625} = 0.9$$ #Step 4: Interpret the result#

The result \(f(0.75)=0.9\) means that when the server has a 75% chance to win any given point in a tennis match, the probability of the server winning a game when serving from deuce is 90%. #b. Evaluate \(f(0.25)\) and interpret the result# #Step 1: Plug in the value of p into the function#

In this step, we will substitute \(p=0.25\) into the function \(f(p)=\frac{p^{2}}{1-2 p(1-p)}\) to get the probability of the server winning a game when serving from deuce. #Step 2: Calculate the probability of winning from deuce#

After substituting \(p=0.25\) into the function \(f(p)\), we can calculate the probability: $$f(0.25)=\frac{(0.25)^2}{1-2(0.25)(1-0.25)}$$ #Step 3: Simplify the expression#

Now, we will simplify the expression to get the final probability value: $$f(0.25)=\frac{0.0625}{1-0.375} = \frac{0.0625}{0.625} = 0.1$$ #Step 4: Interpret the result#

The result \(f(0.25)=0.1\) means that when the server has a 25% chance to win any given point in a tennis match, the probability of the server winning a game when serving from deuce is 10%.

Key concepts

Evaluating Probability Functions

When we talk about evaluating probability functions in the context of sports, and tennis in particular, we're essentially calculating the chances of certain outcomes occurring based on known probabilities. In tennis, for example, we might have a constant probability, denoted as \(p\), representing the server's likelihood of winning any given point. This probability is a number between 0 and 1, where 0 means no chance of winning, and 1 means a certain win.

The function \(f(p)=\frac{p^{2}}{1-2 p(1-p)}\) represents the probability of a player winning a game from deuce, which is a critical point where the score is tied and the outcome can swing in either direction. To evaluate this function, one substitutes the value of \(p\) and simplifies the expression to uncover the likelihood of a victory from that point.

For a highly skilled server with a 75% (0.75) chance of winning a point, the calculation shows a 90% probability of winning the game from deuce with \(f(0.75)=0.9\). This serves as a testament to the player's skills and has implications for match strategies and predictions.

Probability and Sports

Probability plays a significant role in sports analysis, where it provides insights into the expected outcomes of games, matches, and tournaments. In the case of tennis, probabilities help us understand how a server's effectiveness influences their chances of winning games and, by extension, sets and matches. The deuce scenario in tennis is emblematic of this, as it's a situation where the next couple of points can decide the game.

Importance of the Server's Advantage

Being able to quantify the server's advantage provides valuable information. For instance, if we know a server wins 75% of their points, the model tells us they are highly likely to win the game from deuce. Conversely, if their probability of winning a point is only 25%, their chances drop dramatically. These probabilities can inform coaching decisions, like focusing on improving the serve or return game, and can influence betting markets as well as fan expectations.

Interpreting Mathematical Results

Interpreting mathematical results, such as the ones derived from probability functions, involves translating complex numeric data into comprehensible conclusions that can inform decisions. After calculating the probabilities for different scenarios using the function \(f(p)\), one must comprehend what these numbers indicate about the possible outcomes of the tennis game.

For high probability values, the interpretation is that the server has a strong advantage and is likely to win from deuce. In contrast, low values suggest a disadvantage, indicating the need for improvement or a change in strategy. Moreover, these interpretations need to be communicated effectively to non-experts, such as athletes or fans who may not have a deep understanding of the underlying mathematics, making accessibility a key component in discussions about probability in sports.