Past records indicate that the probability of online retail orders that turn out to be fraudulent is 0.06. Suppose​ that, on a given​ day, 21 online retail orders are placed. Assume that the number of online retail orders that turn out to be fraudulent is distributed as a binomial random variable. Complete parts​ (a) through​ (d) below.
Question content area bottom
Part 1
a. What are the mean and standard deviation of the number of online retail orders that turn out to be​ fraudulent?
The mean number of online retail orders that turn out to be fraudulent is

1.2600.
​(Type an integer or a decimal. Round to four decimal places as​ needed.)
Part 2
The standard deviation of the number of fraudulent retail orders is

1.088.
​(Type an integer or a decimal. Round to three decimal places as​ needed.)
Part 3
b. What is the probability that zero online retail orders will turn out to be​ fraudulent?

0.2727 ​(Type an integer or a decimal. Round to four decimal places as​ needed.)
Part 4
c. What is the probability that one online retail order will turn out to be​ fraudulent?

0.3339 ​(Type an integer or a decimal. Round to four decimal places as​ needed.)

To solve the problem, we need to calculate the mean, standard deviation, and probabilities using the binomial distribution. Let's break down each part.

### Part (a): Mean and Standard Deviation

For a binomial distribution:
- The mean (\(\mu\)) is given by:
  \[
  \mu = n \cdot p
  \]
- The standard deviation (\(\sigma\)) is given by:
  \[
  \sigma = \sqrt{n \cdot p \cdot (1 - p)}
  \]

Where:
- \(n\) is the number of trials (21 orders),
- \(p\) is the probability of success (0.06).

**Mean Calculation:**
\[
\mu = 21 \cdot 0.06 = 1.26
\]
So, the mean number of online retail orders that turn out to be fraudulent is **1.2600**.

**Standard Deviation Calculation:**
\[
\sigma = \sqrt{21 \cdot 0.06 \cdot (1 - 0.06)} = \sqrt{21 \cdot 0.06 \cdot 0.94} \approx \sqrt{1.1868} \approx 1.0884
\]
So, the standard deviation of the number of fraudulent retail orders is **1.088** (rounded to three decimal places).

### Part (b): Probability that Zero Online Retail Orders Will Turn Out to Be Fraudulent

We use the binomial probability formula:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
For \(k = 0\):
\[
P(X = 0) = \binom{21}{0} (0.06)^0 (1 - 0.06)^{21} = 1 \cdot 1 \cdot (0.94)^{21} \approx 0.2727
\]
So, the probability that zero online retail orders will turn out to be fraudulent is **0.2727**.

### Part (c): Probability that One Online Retail Order Will Turn Out to Be Fraudulent

For \(k = 1\):
\[
P(X = 1) = \binom{21}{1} (0.06)^1 (1 - 0.06)^{20} = 21 \cdot 0.06 \cdot (0.94)^{20} \approx 0.3339
\]
So, the probability that one online retail order will turn out to be fraudulent is **0.3339**.

### Summary
- **Mean**: 1.2600
- **Standard Deviation**: 1.088
- **Probability of Zero Fraudulent Orders**: 0.2727
- **Probability of One Fraudulent Order**: 0.3339

Would you like any more details or explanations on these calculations?

Here are five related questions to further your understanding:
1. What is the probability of having exactly two fraudulent orders?
2. How would the mean and standard deviation change if the probability of fraud increased to 0.10?
3. What is the cumulative probability of having two or fewer fraudulent orders?
4. How would the results differ if there were 50 orders instead of 21?
5. Can you calculate the probability of having more than three fraudulent orders?

**Tip:** When using the binomial distribution, remember that it applies to situations where there are only two possible outcomes (success or failure) in each trial, and the trials are independent of each other.

Explore how to calculate the mean, standard deviation, and probabilities using the binomial distribution. This problem involves determining these statistics for online retail orders that may be fraudulent, based on a given probability. Suitable for advanced high school students studying probability and statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation