In a particular retail clothing store, approximately 20% of potential customers who walk into the store make a purchase.

In a random sample of 20 customers that walked into the store,
a. what is the probability that exactly 10 of the customers make purchases?
Round to 3 significant digits.
b. what is the probability that at least 2 of the customers make purchases?
Round to 3 significant digits.
c. how many of the customers are expected to make purchases?
Round to 3 significant digits.
d. what is the standard deviation of the number of customers who make purchases?
Round to 3 significant digits.

This problem can be solved using concepts from **binomial distribution**. Here's a quick overview of the key points needed for the solution:

- The binomial distribution is used because we have a fixed number of trials (20 customers), each of which results in one of two outcomes (make a purchase or not).
- The probability of success (a customer making a purchase) is \( p = 0.20 \), and the probability of failure (not making a purchase) is \( 1 - p = 0.80 \).
- The number of trials (customers) is \( n = 20 \).

The probability mass function of a binomial distribution is given by:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}
\]
where:
- \( X \) is the number of customers who make purchases,
- \( n \) is the number of trials (customers),
- \( p \) is the probability of success (making a purchase),
- \( k \) is the number of successes (customers making purchases),
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).

Let's now solve each part of the problem:

### Part (a): Probability that exactly 10 customers make purchases
We need to calculate \( P(X = 10) \). Using the binomial formula:

\[
P(X = 10) = \binom{20}{10} (0.20)^{10} (0.80)^{10}
\]

### Part (b): Probability that at least 2 customers make purchases
This can be expressed as:

\[
P(X \geq 2) = 1 - P(X = 0) - P(X = 1)
\]

We'll calculate \( P(X = 0) \) and \( P(X = 1) \) using the binomial distribution formula, and then subtract these values from 1.

### Part (c): Expected number of customers who make purchases
The expected value \( E(X) \) of a binomial distribution is given by:

\[
E(X) = n \cdot p = 20 \cdot 0.20
\]

### Part (d): Standard deviation of the number of customers who make purchases
The standard deviation \( \sigma \) of a binomial distribution is given by:

\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{20 \cdot 0.20 \cdot 0.80}
\]

Let's calculate these values.Here are the solutions to the problem:

### (a) Probability that exactly 10 customers make purchases:
\[
P(X = 10) = 0.002
\]

### (b) Probability that at least 2 customers make purchases:
\[
P(X \geq 2) = 0.931
\]

### (c) Expected number of customers who make purchases:
\[
E(X) = 4.0
\]

### (d) Standard deviation of the number of customers who make purchases:
\[
\sigma = 1.789
\]

Let me know if you need any clarifications or further details.

Here are 5 related questions for practice:
1. What is the probability that exactly 5 customers make purchases?
2. What is the probability that fewer than 3 customers make purchases?
3. What is the cumulative probability that 15 or more customers make purchases?
4. How would the expected value change if the probability of making a purchase was 25%?
5. How does the variance of the distribution change if the sample size increases to 50 customers?

**Tip:** In binomial distribution, the expected value tells us the average number of successes, while the standard deviation provides a measure of how much the number of successes might vary.

Explore how to calculate the probability of customer purchases in a retail store using binomial distribution. Solve for the probability of exactly 10 purchases, the probability of at least 2 purchases, the expected number of customers making purchases, and the standard deviation. This problem applies key concepts from probability theory and binomial distribution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation