In a typical month, an insurance agent presents life insurance plans to 40 potential customers. Historically, one in four such customers chooses to buy life insurance from this agent. What is the probability that exactly 6 customers will buy life insurance from this agent in the coming month? 0.0530 0.1057 0.0282 0.1444 0.1397

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### Problem Statement:

In a typical month, an insurance agent presents life insurance plans to 40 potential customers. Historically, one in four such customers chooses to buy life insurance from this agent. What is the probability that exactly 6 customers will buy life insurance from this agent in the coming month?

### Possible Answers:
- 0.0530
- 0.1057
- 0.0282
- 0.1444
- 0.1397

### Explanation:

This question involves calculating the probability of a specific number of events (in this case, 6) occurring out of a fixed number of trials (40), where each trial has two possible outcomes (customer buys insurance or does not buy insurance) and the probability of the outcomes is known (1 out of 4).

This is a binomial probability problem, where the number of trials \(n = 40\) and the probability of success in each trial \(p = 0.25\). The probability of exactly \(k = 6\) successes can be calculated using the binomial probability formula:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Where:
- \(\binom{n}{k}\) is the binomial coefficient which can be calculated as \(\frac{n!}{k!(n-k)!}\).
- \(p\) is the probability of a single success.
- \(1-p\) is the probability of a single failure.

### Detailed Calculation:
For \(n = 40\), \(k = 6\), and \(p = 0.25\):

\[ \binom{40}{6} = \frac{40!}{6!(40-6)!} \]
\[ p^k = (0.25)^6 \]
\[ (1-p)^{n-k} = (0.75)^{34} \]

\[ P(X = 6) = \binom{40}{6} \times (0.25)^6 \times (0.75)^{34} \]

This problem can be solved manually or using a statistical calculator or software to get the precise value.
Transcribed Image Text:### Problem Statement: In a typical month, an insurance agent presents life insurance plans to 40 potential customers. Historically, one in four such customers chooses to buy life insurance from this agent. What is the probability that exactly 6 customers will buy life insurance from this agent in the coming month? ### Possible Answers: - 0.0530 - 0.1057 - 0.0282 - 0.1444 - 0.1397 ### Explanation: This question involves calculating the probability of a specific number of events (in this case, 6) occurring out of a fixed number of trials (40), where each trial has two possible outcomes (customer buys insurance or does not buy insurance) and the probability of the outcomes is known (1 out of 4). This is a binomial probability problem, where the number of trials \(n = 40\) and the probability of success in each trial \(p = 0.25\). The probability of exactly \(k = 6\) successes can be calculated using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where: - \(\binom{n}{k}\) is the binomial coefficient which can be calculated as \(\frac{n!}{k!(n-k)!}\). - \(p\) is the probability of a single success. - \(1-p\) is the probability of a single failure. ### Detailed Calculation: For \(n = 40\), \(k = 6\), and \(p = 0.25\): \[ \binom{40}{6} = \frac{40!}{6!(40-6)!} \] \[ p^k = (0.25)^6 \] \[ (1-p)^{n-k} = (0.75)^{34} \] \[ P(X = 6) = \binom{40}{6} \times (0.25)^6 \times (0.75)^{34} \] This problem can be solved manually or using a statistical calculator or software to get the precise value.
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