QuestionJuly 15, 2025

10. In the NBA in 2020, there were an average of 4.875 personal fouls per quarter. What is the probability that a randomly selected game quarter had more than 7 fouls? (10 points)

10. In the NBA in 2020, there were an average of 4.875 personal fouls per quarter. What is the probability that a randomly selected game quarter had more than 7 fouls? (10 points)
10. In the NBA in 2020, there were an average of 4.875 personal fouls per quarter. What is the probability that a randomly selected game quarter had more than 7 fouls? (10 points)

Solution
4.2(224 votes)

Answer

Approximately 0.172 Explanation 1. Define the Distribution The number of fouls per quarter follows a Poisson distribution with \lambda = 4.875. 2. Calculate Probability for More Than 7 Fouls Use the cumulative distribution function (CDF) for Poisson to find P(X \leq 7) and subtract from 1: P(X > 7) = 1 - P(X \leq 7). 3. Compute CDF for X \leq 7 P(X \leq 7) = e^{-\lambda} \sum_{k=0}^{7} \frac{\lambda^k}{k!} where \lambda = 4.875. 4. Calculate P(X > 7) Substitute values: P(X > 7) = 1 - (e^{-4.875} \sum_{k=0}^{7} \frac{4.875^k}{k!}).

Explanation

1. Define the Distribution<br /> The number of fouls per quarter follows a Poisson distribution with $\lambda = 4.875$.<br /><br />2. Calculate Probability for More Than 7 Fouls<br /> Use the cumulative distribution function (CDF) for Poisson to find $P(X \leq 7)$ and subtract from 1: <br /> $P(X > 7) = 1 - P(X \leq 7)$.<br /><br />3. Compute CDF for $X \leq 7$<br /> $P(X \leq 7) = e^{-\lambda} \sum_{k=0}^{7} \frac{\lambda^k}{k!}$ where $\lambda = 4.875$.<br /><br />4. Calculate $P(X > 7)$<br /> Substitute values: $P(X > 7) = 1 - (e^{-4.875} \sum_{k=0}^{7} \frac{4.875^k}{k!})$.
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