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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblem 4.2.22
Chapter 4, Problem 4.2.22

Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.


Penalty Kicks Argentine soccer player Lionel Messi converts 78% of his penalty kicks. Suppose Messi takes six penalty kicks next season. Find the probability that the number he converts is (a) exactly six, (b) at most three, and (c) more than three. (Source: Transfermarkt)

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Step 1: Recognize that this is a binomial probability problem. The binomial distribution is used when there are a fixed number of independent trials (n), each with two possible outcomes (success or failure), and the probability of success (p) is constant. Here, n = 6 (penalty kicks), p = 0.78 (probability of converting a penalty kick), and q = 1 - p = 0.22 (probability of missing a penalty kick).
Step 2: Use the binomial probability formula to calculate the probability of exactly six successes (part a). The formula is: P(X = k) = (n choose k) * p^k * q^(n-k), where 'k' is the number of successes. For part (a), substitute k = 6, n = 6, p = 0.78, and q = 0.22 into the formula.
Step 3: For part (b), calculate the probability of at most three successes (P(X ≤ 3)). This is the sum of the probabilities for X = 0, X = 1, X = 2, and X = 3. Use the binomial probability formula for each value of k (0 through 3) and sum the results: P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).
Step 4: For part (c), calculate the probability of more than three successes (P(X > 3)). This is the complement of P(X ≤ 3), so use the formula: P(X > 3) = 1 - P(X ≤ 3). Use the result from part (b) to find this value.
Step 5: If convenient, use technology (e.g., a statistical calculator, spreadsheet software, or statistical software) to compute the binomial probabilities for each part. Alternatively, use Table 2 in Appendix B if it provides cumulative binomial probabilities for the given parameters.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, Messi's penalty kicks represent the trials, where a success is defined as converting a kick. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).
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Probability Mass Function (PMF)

The probability mass function for a binomial distribution gives the probability of obtaining exactly k successes in n trials. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k). This function is essential for determining the probabilities of Messi converting a specific number of penalty kicks, such as exactly six or at most three.
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Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain number. For the given problem, calculating the probability of Messi converting at most three kicks involves summing the probabilities of converting zero, one, two, or three kicks. This concept is crucial for answering parts (b) and (c) of the question.
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Related Practice
Textbook Question

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