Suppose a life insurance company sells a 

​$290 comma 000290,000 

​1-year term life insurance policy to a 

2020​-year-old 

female for 

​$350350. 

According to the National Vital Statistics​ Report, 58(21), the probability that the female survives the year is 

0.9995440.999544. 

The expected value of this policy to the insurance company is 

​$217.76217.76. 

What is the standard deviation of the value of the life insurance​ policy? Why is the value so​ high?

To calculate the **standard deviation** of the value of the life insurance policy, we first need to recognize that this is a problem involving a **binomial distribution** since the outcome involves two possibilities: either the policyholder survives (with a probability of 0.999544) or dies (with a probability of 1 - 0.999544 = 0.000456). 

Here’s the step-by-step process:

### 1. Calculate the Expected Value (already provided):
The expected value $ E(X) $ of the policy to the insurance company is already given as $217.76.

### 2. Define the possible outcomes and probabilities:
- If the policyholder survives (with probability 0.999544), the insurance company gains the premium of $350.
- If the policyholder dies (with probability 0.000456), the insurance company loses the payout amount of $290,000 minus the premium of $350. The net loss is $ -290,000 + 350 = -289,650 $.

### 3. Calculate the variance $ \text{Var}(X) $:
For a binomial distribution, the variance formula is:

\[
\text{Var}(X) = p \times (1-p) \times (x_1 - x_2)^2
\]

where:
- $ p = 0.999544 $ is the probability that the policyholder survives,
- $ 1 - p = 0.000456 $ is the probability that the policyholder dies,
- $ x_1 = 350 $ is the gain if the policyholder survives,
- $ x_2 = -289,650 $ is the loss if the policyholder dies.

Let’s compute the variance step-by-step.

### 4. Calculate the standard deviation:
Once the variance is computed, the standard deviation $ \sigma(X) $ is the square root of the variance.

Now let's compute it.The **standard deviation** of the value of the life insurance policy is approximately **$6,182.41**.

### Why is the standard deviation so high?
The standard deviation is high because of the **large potential loss** ($289,650) in the case of death. Although the probability of death is small (0.000456), the size of the loss is significant, which contributes to a large variance and consequently a high standard deviation. This reflects the high level of risk associated with the insurance policy.

Would you like further details on any part of the solution?

### Related questions:
1. How is the expected value calculated in a binomial distribution?
2. Why is the variance of a binomial distribution linked to the size of the potential loss?
3. How would the standard deviation change if the payout amount were different?
4. What is the relationship between risk and standard deviation in insurance policies?
5. How do insurance companies use probability to price policies?

### Tip:
Insurance companies typically balance premium amounts with the risk (measured by probability and variance) to ensure profitability across many policies.

Suppose a life insurance company sells a $290,000 1-year term life insurance policy to a 20-year-old female for $350. According to the National Vital Statistics Report, the probability that the female survives the year is 0.999544. The expected value of this policy to the insurance company is $217.76. What is the standard deviation of the value of the life insurance policy? Why is the value so high?

Learn how to calculate the standard deviation of a life insurance policy sold to a 20-year-old female with a 0.999544 probability of surviving the year. Explore key concepts like expected value, binomial distribution, and the relationship between large potential losses and high standard deviation. Suitable for advanced high school or college-level students.

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