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Chapter 5: Problem 88

Winning at tennis A player serving in tennis has two chances to get a serve into play. If the first serve goes out of bounds, the player serves again. If the second serve is also out, the player loses the point. Here are probabilities based on four years of the Wimbledon Championship:18 $$ \begin{array}{c}{P(\text { lst serve in })=0.59 \quad P(\text { win point } | \text { lst serve in })=0.73} \\ {\qquad P(2 \text { nd serve in } | \text { lst serve out })=0.86} \\ {P(\text { win point } | \text { lst serve out and } 2 \text { nd serve in })=0.59}\end{array} $$ (a) Make a tree diagram for the results of the two serves and the outcome (win or lose) of the point. (b) What is the probability that the serving player wins the point? Show your work.

Short Answer

Expert verified
The probability of winning the point is 0.6393.

Step by step solution

01

Interpret Given Probabilities

We are given several conditional probabilities about a tennis player's serves at Wimbledon. - Probability that the first serve is in: \(P(\text{1st serve in})=0.59\). - If the first serve is in, probability of winning the point: \(P(\text{win point} | \text{1st serve in})=0.73\).- Probability of landing the second serve in if the first serve is out: \(P(2\text{nd serve in} | \text{1st serve out})=0.86\).- If the first serve is out and the second serve is in, probability of winning the point: \(P(\text{win point} | \text{1st serve out and 2nd serve in})=0.59\).
02

Sketch the Tree Diagram

Draw a tree diagram showing all possible outcomes: 1. The first split shows the 1st serve being in or out. 2. If the 1st serve is in, the outcome is either winning or losing the point. 3. If the 1st serve is out, draw the next split for the 2nd serve being in or out. 4. If the 2nd serve is in, the outcome is either winning or losing the point. 5. If the 2nd serve is out, the player automatically loses the point.
03

Calculate Probabilities for Each Path

The probability of each path can be calculated by multiplying the probability of events along the path:1. Win by 1st serve in: \(P(\text{1st serve in} \text{ and win}) = P(\text{1st serve in}) \times P(\text{win point} | \text{1st serve in}) = 0.59 \times 0.73 = 0.4307\).2. Lose by 1st serve in: \(P(\text{1st serve in} \text{ and lose}) = 0.59 \times (1-0.73) = 0.59 \times 0.27 = 0.1593\).3. 1st serve out and win by 2nd serve in: \(P(\text{1st serve out and 2nd serve in and win}) = (1-0.59) \times 0.86 \times 0.59 = 0.41 \times 0.86 \times 0.59 = 0.2086\).4. 1st serve out, 2nd serve in, and lose: \(P(\text{1st serve out and 2nd serve in and lose}) = 0.41 \times 0.86 \times (1-0.59) = 0.41 \times 0.86 \times 0.41 = 0.1449\).5. Automatic loss by 2nd serve out: \(P(\text{1st serve out and 2nd serve out}) = 0.41 \times (1-0.86) = 0.41 \times 0.14 = 0.0574\).
04

Total Probability of Winning

The probability that the player wins the point is the sum of the probabilities for all paths that result in a win:- \(P(\text{win}) = P(\text{1st serve in and win}) + P(\text{1st serve out and 2nd serve in and win}) = 0.4307 + 0.2086 = 0.6393\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Conditional probability is a concept that helps us to determine the likelihood of an event occurring, given that another event has already occurred. It's crucial in understanding how probabilities change with new information. For instance, in the context of tennis, knowing the first serve is in gives different probabilities than if the serve was out.

We denote conditional probability with the notation \(P(A|B)\), meaning "the probability of A occurring given that B has occurred."
  • If the first serve is in, the probability of winning the point \(P(\text{win} | \text{1st serve in})\) is 0.73.
  • If the first serve is out, the player relies on the second serve, with \(P(\text{2nd serve in} | \text{1st serve out}) = 0.86\).
This reflects a common situation in sports statistics where conditional probabilities are calculated to fine-tune predictions and strategies.
Tree Diagram
Representing a series of events and their associated probabilities visually can be greatly beneficial, especially for complex scenarios. A tree diagram is a graphical tool used to map out all possible outcomes of a set of events, showing their probabilities along each branch.

In our tennis scenario:
  • The first decision point (or "node") is whether the first serve is in or out.
  • Each subsequent node represents the next possible outcome (e.g., if the 1st serve is out, then whether the 2nd serve is in or out).
  • The endpoints of the tree ("leaves") show the final outcomes: win or lose the point.
Using a tree diagram, you can easily visualize all the paths through the nodes, helping you track the sequence of events and their cumulative probabilities. In this example, you can quickly determine the probability of the serving player winning or losing as the events unfold.
Statistical Analysis
Statistical analysis involves calculating probabilities in order to make informed predictions or decisions based on data. It involves a thorough process of interpreting data, calculating outcomes and deriving insights.

In the tennis example, we begin by calculating the probabilities of different serve outcomes:
  • First, we determine the likelihood of the 1st serve being in and winning the point.
  • Then, we calculate scenarios where the 1st serve is out and the 2nd is in, including both winning and losing outcomes.
By multiplying these conditional probabilities, we get the joint probabilities for each path. Finally, the probability of winning the point is the total of all paths leading to a win.
  • Total probability of winning: \(0.4307 \text{ (1st in and win)} + 0.2086 \text{ (2nd in and win if 1st out)} = 0.6393\).
Such analysis not only helps in understanding possible outcomes but also optimizes strategies based on which serves and subsequent plays are statistically more beneficial.

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