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Chapter 19: Problem 47

Alysha makes \(40 \%\) of her free throws. She takes five free throws in a game. If the shots are independent of each other, the probability that she makes exactly one of five shots is about (a) \(0.259\). (b) \(0.115 .\) (c) \(0.052\). (d) \(0.200\).

Short Answer

Expert verified
The probability that Alysha makes exactly one of five shots is approximately 0.259, which is option (a).

Step by step solution

01

Understand the Problem

We need to find the probability that Alysha makes exactly one successful free throw out of five attempts, where each shot has a 40% chance of success, and the shots are independent events.
02

Define the Variables

Let the probability of making a shot be \( p = 0.4 \) and the probability of missing a shot be \( q = 0.6 \). We are dealing with a binomial distribution scenario with \( n = 5 \) trials.
03

Binomial Probability Formula

The probability of getting exactly \( k \) successes in \( n \) independent Bernoulli trials is given by the binomial formula: \[ P(X = k) = \binom{n}{k} \cdot p^k \cdot q^{n-k} \] where \( \binom{n}{k} \) is the binomial coefficient.
04

Calculate the Binomial Coefficient

For \( k = 1 \) and \( n = 5 \), the binomial coefficient \( \binom{5}{1} \) is calculated as follows:\( \binom{5}{1} = \frac{5!}{1!(5-1)!} = 5 \).
05

Calculate Probability of Exactly One Success

Plug \( n = 5 \), \( k = 1 \), \( p = 0.4 \), and \( q = 0.6 \) into the formula:\[ P(X = 1) = \binom{5}{1} \cdot (0.4)^1 \cdot (0.6)^{5-1} = 5 \cdot 0.4 \cdot 0.1296 \] This gives: \[ P(X = 1) = 5 \cdot 0.05184 = 0.2592 \]
06

Select the Closest Option

After calculating, the closest probability value from the given options is \( 0.259 \), which corresponds to option (a).

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Key Concepts

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

Probability Calculation
Probability calculation involves determining the likelihood of a specific outcome occurring from a set of possible outcomes. In the context of binomial distribution, probability calculation becomes essential to predict the outcome of a series of independent events, like Alysha's free throws. Alysha's problem involves calculating the probability that she makes exactly one shot out of five attempts.
To solve this, we use the probability values: the probability of a successful shot, and the probability of a missed shot:
  • The probability of making a shot ( p ) is 0.4.
  • The probability of missing a shot ( q ) is the complement, 0.6 (since 1 - 0.4 = 0.6 ).
These probabilities are applied in a binomial probability formula to find exactly how many successes (or made shots) occur among a fixed number of independent trials.
Bernoulli Trials
Bernoulli trials are the foundation of many probability models. Each Bernoulli trial is an experiment or a process that results in a binary outcome: success or failure. In Alysha's exercise, each free throw she attempts is a Bernoulli trial.
Understanding Bernoulli trials is crucial for calculating probabilities using the binomial distribution. Here, we apply it as follows: - Alysha takes five free throws, each considered a separate trial. - Each shot has a consistent probability of success (making the shot) of 0.4. The concept of independent trials means that whatever happens on one trial does not affect the outcomes of others. This assumption of independence is crucial. For Alysha, if her chance remains the same throughout each shot, it confirms this idea and allows us to use binomial distribution naturally.
Binomial Coefficient
The binomial coefficient is a key component in the calculation of probabilities using the binomial distribution formula. This coefficient is represented as \( \binom{n}{k} \) , where \( n \) is the total number of trials, and \( k \) is the number of successful outcomes we aim to calculate. For Alysha's scenario, we need to find the probability of exactly one success.
The binomial coefficient is calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] In this case,
  • \( n = 5 \) (five shots)
  • \( k = 1 \) (one successful shot)
The computation becomes: \[ \binom{5}{1} = \frac{5!}{1!(5-1)!} = 5 \] This coefficient multiplies with the power of success probability ( p^k ) and failure probability ( q^{n-k} ) in the binomial formula to yield the total probability of getting exactly one success.

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Most popular questions from this chapter

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