Questions: Insurance: An insurance company sells a 1-year term life insurance policy to an 84 -year-old man. The man pays a premium of 2500. If he dies within 1 year, the company will pay 36,000 to his beneficiary. According to the U.S. Centers for Disease Control and Prevention, the probability that an 84 -year-old man will be alive 1 year later is 0.9318 . Let x be the profit made by the insurance company. (a) Find the probability distribution. The probability distribution is x, P(x), 2500

Solution Steps

To find the probability distribution, we need to consider two scenarios: the man survives the year or the man dies within the year. We will calculate the profit for each scenario and their respective probabilities.

1. If the man survives, the insurance company keeps the premium of $2500.
2. If the man dies, the insurance company pays out $36000, resulting in a loss of $36000 - $2500 = $33500.

The probability of the man surviving is given as 0.9318, and the probability of the man dying is 1 - 0.9318.

1. Calculate the profit for each scenario.
2. Determine the probability for each scenario.
3. Construct the probability distribution table.

We have two scenarios for the insurance company based on whether the 84-year-old man survives or dies within the year.

1. If the man survives, the profit \( x \) is given by: \[ x = 2500 \]

2. If the man dies, the profit \( x \) is calculated as: \[ x = 2500 - 36000 = -33500 \]

The probabilities associated with each scenario are as follows:

1. The probability that the man survives is: \[ P(x = 2500) = 0.9318 \]

2. The probability that the man dies is: \[ P(x = -33500) = 1 - 0.9318 = 0.0682 \]

We can summarize the profit and their corresponding probabilities in a probability distribution table:

\[ \begin{array}{|c|c|c|} \hline x & P(x) \\ \hline 2500 & 0.9318 \\ \hline -33500 & 0.0682 \\ \hline \end{array} \]

Final Answer