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An insurance company sells a 1-year term life insurance policy to an 81-year-old woman. The woman pays a premium of $1500. If she dies within 1 year, the company will pay $36,000 to her beneficiary. According to the U.S. Centers for Disease Control and Prevention, the probability that an 81-year-old woman will be alive 1 year later is 0.9605. Let X be the profit made by the insurance company.

Find the probability distribution.
x| [?] 1500
P(x)| [?] [?]

7 answers

To find the probability distribution for the profit \(X\) made by the insurance company from selling this term life insurance policy, we need to calculate the possible outcomes for \(X\) based on the two scenarios: whether the woman survives or dies within the year.

  1. Scenario 1: The woman survives (with a probability of \(P(\text{survive}) = 0.9605\)):

    • In this case, the insurance company keeps the premium and does not have to pay the death benefit.
    • The profit, \(X\), in this case would be: \[ X = \text{premium} = 1500 \]

  2. Scenario 2: The woman dies (with a probability of \(P(\text{die}) = 1 - P(\text{survive}) = 1 - 0.9605 = 0.0395\)):

    • In this case, the insurance company pays out the death benefit of $36,000.
    • The profit, \(X\), in this scenario would be: \[ X = \text{premium} - \text{death benefit} = 1500 - 36000 = -34500 \]

We can summarize the probability distribution of the profit \(X\) as follows:

\[ \begin{array}{|c|c|} \hline x & P(X = x) \ \hline 1500 & 0.9605 \ -34500 & 0.0395 \ \hline \end{array} \]

Conclusion

  • The probability distribution of the profit \(X\) for the insurance company is:
    • \(X = 1500\) with probability \(0.9605\)
    • \(X = -34500\) with probability \(0.0395\)
(b) Find the expected value of the profit.

To find the expected value of the profit \(E(X)\), we use the formula for expected value given a discrete probability distribution:

\[ E(X) = \sum x P(X = x) \]

From the probability distribution we established:

\[ \begin{array}{|c|c|} \hline x & P(X = x) \ \hline 1500 & 0.9605 \ -34500 & 0.0395 \ \hline \end{array} \]

Now we can compute the expected value:

\[ E(X) = (1500 \times 0.9605) + (-34500 \times 0.0395) \]

Calculating each term:

  1. For the first term: \[ 1500 \times 0.9605 = 1440.75 \]

  2. For the second term: \[ -34500 \times 0.0395 = -1365.75 \]

Now add the two results together:

\[ E(X) = 1440.75 - 1365.75 = 75 \]

Conclusion

The expected value of the profit \(E(X)\) for the insurance company is $75.

75 is not correct

Let’s carefully recalculate the expected value of the profit \(E(X)\) given the previously established profit values and their respective probabilities.

The profit values and probabilities are:

\[ \begin{array}{|c|c|} \hline x & P(X = x) \ \hline 1500 & 0.9605 \ -34500 & 0.0395 \ \hline \end{array} \]

To find the expected value of \(X\), we use:

\[ E(X) = (1500 \times 0.9605) + (-34500 \times 0.0395) \]

Now let's calculate each term carefully:

  1. Calculate the first term: \[ 1500 \times 0.9605 = 1440.75 \]

  2. Calculate the second term: \[ -34500 \times 0.0395 = -1365.75 \]

  3. Add both terms together: \[ E(X) = 1440.75 - 1365.75 = 75 \]

The calculations are indeed correct, and the expected value of the profit for the insurance company is $75.

If you still believe there is a discrepancy, could you please specify your concerns or the values you have in mind? That way, we can clarify any possible misunderstandings.

The answer is 78

Let’s carefully recheck the expectations calculation step by step.

We have two scenarios for the profit \(X\) based on whether the woman lives or dies:

  1. If the woman survives (with probability \(P(\text{survive}) = 0.9605\)):

    • Profit \(X = 1500\)

  2. If the woman dies (with probability \(P(\text{die}) = 1 - 0.9605 = 0.0395\)):

    • Profit \(X = 1500 - 36000 = -34500\)

Next, we calculate \(E(X)\):

\[ E(X) = (1500 \times 0.9605) + (-34500 \times 0.0395) \]

Now, calculate each product:

  1. Profit if the woman survives: \[ 1500 \times 0.9605 = 1440.75 \]

  2. Profit if the woman dies: \[ -34500 \times 0.0395 = -1365.75 \]

Now, combine these results:

\[ E(X) = 1440.75 - 1365.75 \]

Calculating that gives:

\[ E(X) = 75 \]

Double-checking Probabilities

The probability of each event seems accurate:

  • Probability of survival: \(0.9605\)
  • Probability of dying: \(0.0395\)

Re-evaluating the Values

To reach your expected value of \(78\):

Make sure we did not overlook any crucial detail. Let's check all steps again.

  1. Profit From Surviving: \[ 1500 \times 0.9605 = 1440.75 \]

  2. Profit From Dying: \[ -34500 \times 0.0395 = -1365.75 \]

Adding them:

\[ E(X) = 1440.75 - 1365.75 = 75 \]

Conclusion

From this consistent calculation across each step, the expected value \(E(X)\) remains \(75\). If you believe the answer should be \(78\), please verify the parameters or provide any additional context. However, based on the calculations performed, I confirm that the answer is indeed \(75\).