Math Problem Statement
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?
0.2653 (Type an integer or a decimal. Round to four decimal places as needed.)
Solution
Let's walk through the problem step by step.
Part (a): Mean and Standard Deviation
Given:
- Probability of a fraudulent order,
- Number of orders,
The number of fraudulent orders follows a binomial distribution, .
Mean: The mean of a binomial distribution is given by: Substituting the values: So, the mean is 1.2600.
Standard Deviation: The standard deviation of a binomial distribution is given by: Substituting the values: So, the standard deviation is 1.088.
Part (b): Probability that zero orders are fraudulent
We need to find the probability that where is the number of fraudulent orders.
The probability mass function for a binomial distribution is: For : So, the probability that zero orders are fraudulent is 0.2653.
Summary:
- The mean number of fraudulent orders is 1.2600.
- The standard deviation of the number of fraudulent orders is 1.088.
- The probability that zero orders are fraudulent is 0.2653.
Would you like further details or have any other questions?
Here are some additional questions you might consider:
- What is the probability of exactly 1 fraudulent order?
- What is the cumulative probability of having 2 or fewer fraudulent orders?
- How does the probability change if the probability of fraud increases to 0.10?
- What is the expected value and standard deviation if 50 orders are placed instead of 21?
- How would you calculate the probability of having more than 3 fraudulent orders?
Tip: Understanding the binomial distribution is essential for modeling scenarios where there are fixed numbers of independent trials, each with the same probability of success.
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Math Problem Analysis
Mathematical Concepts
Binomial Distribution
Probability Theory
Formulas
Mean of Binomial Distribution
Standard Deviation of Binomial Distribution
Theorems
-
Suitable Grade Level
Advanced High School
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