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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
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Tutti i libri di testoTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblema 5.CRE.3c
Capitolo 5, Problema 5.CRE.3c

Tennis Challenge In a recent U.S. Open tennis tournament, there were 945 challenges made by singles players, and 255 of them resulted in referee calls that were overturned. The accompanying table lists the results by gender.


Table showing tennis challenge results: Men upheld 160, rejected 398; Women upheld 95, rejected 292.


c. If two different challenges are randomly selected without replacement, find the probability that they both resulted in an overturned call.

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Step 1: Identify the total number of challenges that resulted in overturned calls. From the table, challenges upheld with overturned calls are 160 for men and 95 for women. Add these values to find the total overturned calls: 160 + 95.
Step 2: Calculate the probability of selecting one challenge that resulted in an overturned call. Divide the total number of overturned calls by the total number of challenges (945). Use the formula: \( P(A) = \frac{\text{Number of Overturned Calls}}{\text{Total Challenges}} \).
Step 3: Since the challenges are selected without replacement, calculate the probability of selecting a second challenge that resulted in an overturned call after the first one has been selected. Subtract 1 from the total number of overturned calls and subtract 1 from the total number of challenges, then divide: \( P(B|A) = \frac{\text{Number of Overturned Calls - 1}}{\text{Total Challenges - 1}} \).
Step 4: Multiply the probabilities from Step 2 and Step 3 to find the joint probability that both selected challenges resulted in overturned calls. Use the formula for dependent events: \( P(A \cap B) = P(A) \times P(B|A) \).
Step 5: Simplify the expression obtained in Step 4 to get the final probability. Ensure all calculations are consistent with the rules of probability and dependent events.

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Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this context, it involves calculating the chance of selecting two challenges that resulted in overturned calls from a total number of challenges. Understanding how to compute probabilities, especially in scenarios involving combinations or selections without replacement, is crucial for solving the given question.
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Introduction to Probability

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. In this problem, we need to determine how many ways we can select two overturned calls from the total number of overturned calls. This concept is essential for calculating the total number of favorable outcomes when determining the probability of selecting two specific challenges.
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Combinations

Total Outcomes

Total outcomes represent the complete set of possible results in a probability experiment. In this case, it includes all challenges made by players, both overturned and upheld. Knowing the total number of challenges is necessary to calculate the probability of selecting two overturned calls, as it provides the denominator in the probability formula.
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Fundamental Counting Principle
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Tennis Challenge In a recent U.S. Open tennis tournament, there were 945 challenges made by singles players, and 255 of them resulted in referee calls that were overturned. The accompanying table lists the results by gender.



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0 0 1 2 17 28 21 8


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