An insurance company sells a 1-year term life insurance policy to an 85-year-old woman. The woman pays a premium of $100. If she dies within 1 year, the company will pay $83,000 to her beneficiary. According to the U.S. Centers for Disease Control and Prevention, the probability that an 85-year-old woman will be alive 1 year later is 0.9512. Let X be the profit made by the insurance company.

Question

A- Find the probability of distribution
x| [?] 4100
P(x)| [?] [?]

Answer 1

To find the probability distribution of the profit \( X \) for the insurance company, we need to determine the possible outcomes for \( X \) and their corresponding probabilities.

Step 1: Determine the outcomes for \( X \)

1. If the woman survives (with probability \( P(\text{Survive}) = 0.9512 \)):

   • The insurance company keeps the premium of $100.
   • The profit \( X \) in this case is: \[ X = 100 \]

2. If the woman dies (with probability \( P(\text{Die}) = 1 - P(\text{Survive}) = 1 - 0.9512 = 0.0488 \)):

   • The insurance company pays out $83,000 to the beneficiary and keeps the premium of $100.
   • The profit \( X \) in this case is: \[ X = 100 - 83,000 = -82,900 \]

Step 2: Define the probability distribution

Now we can define the probability distribution:

| \( x \) | \( P(X) \) | |----------|----------------------| | 100 | 0.9512 | | -82,900 | 0.0488 |

Summary of the probability distribution

• The profit \( X \) can either be \( 100 \) with a probability of \( 0.9512 \) or \( -82,900 \) with a probability of \( 0.0488 \).

Answer 2

No, the probability distribution is
x= ?
x= 4100
the P(x) is both question marks

Answer 3

Thank you for the clarification. It seems there might have been a misunderstanding regarding the desired profit value. Let's reevaluate the situation based on the revised details provided.

Problem Statement Recap

• Premium paid: $100
• Payout if the woman dies: $83,000
• Probability she survives: \( P(\text{Survive}) = 0.9512 \)
• Probability she dies: \( P(\text{Die}) = 1 - P(\text{Survive}) = 0.0488 \)

Profits for the Insurance Company

1. If the woman survives:

   • Profit = Premium = $100

2. If the woman dies:

   • Profit = Premium - Payout = $100 - $83,000 = -$82,900

Profit Outcomes

• If she survives: Profit \( X = 100 \)
• If she dies: Profit \( X = -82,900 \)

Probability Distribution Summary

The two outcomes of the profit, \( X \), and their respective probabilities are:

• \( x = 100 \) with \( P(X = 100) = 0.9512 \)
• \( x = -82,900 \) with \( P(X = -82,900) = 0.0488 \)

Summary Table

Here's the final output based on your request:

| \( x \) | \( P(x) \) | |---------------|---------------------| | 100 | 0.9512 | | -82,900 | 0.0488 |

If your context involves \( x = 4100 \) specifically, could you clarify its relevance or how it applies to this insurance scenario? Based on the information so far, I do not see a direct relationship to the profit calculations previously mentioned.

Answer 4

Sorry, I was incorrect. The premium she pays is $4100 not 100

Answer 5

No problem! Let’s go through the calculation again with the updated premium of $4,100.

Given Information

• Premium paid: $4,100
• Payout if the woman dies: $83,000
• Probability she survives: \( P(\text{Survive}) = 0.9512 \)
• Probability she dies: \( P(\text{Die}) = 1 - P(\text{Survive}) = 0.0488 \)

Calculate Profit Outcomes

1. If the woman survives:

   • Profit = Premium = \( 4,100 \)

2. If the woman dies:

   • Profit = Premium - Payout = \( 4,100 - 83,000 = -78,900 \)

Probability Distribution

• Outcome if she survives:

  • \( X = 4,100 \) with \( P(X = 4,100) = 0.9512 \)

• Outcome if she dies:

  • \( X = -78,900 \) with \( P(X = -78,900) = 0.0488 \)

Summary Table

Here’s the final probability distribution based on the corrected premium:

| \( x \) | \( P(x) \) | |---------------|---------------------| | \( 4,100 \) | \( 0.9512 \) | | \( -78,900 \) | \( 0.0488 \) |

This table summarizes the profit outcomes for the insurance company and the associated probabilities. Thank you for your patience! If you have any further questions, feel free to ask.

Answer 6

Great! using the probability distribution, what is the expected value of the profit?

Answer 7

To calculate the expected value of the profit \( E(X) \), we use the formula:

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

Where \( x \) represents the profit outcomes and \( P(x) \) represents the corresponding probabilities.

Given Values

1. \( x_1 = 4,100 \) with \( P(x_1) = 0.9512 \)
2. \( x_2 = -78,900 \) with \( P(x_2) = 0.0488 \)

Calculate the Expected Value

\[ E(X) = (4,100 \cdot 0.9512) + (-78,900 \cdot 0.0488) \]

Step 1: Calculate the first term

\[ 4,100 \cdot 0.9512 = 3,903.92 \]

Step 2: Calculate the second term

\[ -78,900 \cdot 0.0488 = -3,844.32 \]

Step 3: Combine the results

\[ E(X) = 3,903.92 - 3,844.32 = 59.60 \]

Conclusion

The expected value of the profit \( E(X) \) for the insurance company is $59.60. This means, on average, the company expects to profit approximately $59.60 per policy sold to an 85-year-old woman, given these probabilities and payouts.

Answer 8

4,100 times 0.9512 is 3,899.92 and -78900 times 0.0488 is -3850.32

Answer 9

Thank you for your patience, and I appreciate your clarification on the calculations! Let's correct the calculations with the provided multipliers:

Given Values Recap

1. First profit outcome:

   • \( x_1 = 4,100 \)
   • \( P(x_1) = 0.9512 \)

2. Second profit outcome:

   • \( x_2 = -78,900 \)
   • \( P(x_2) = 0.0488 \)

Expected Value Calculation

To find the expected value \( E(X) \):

\[ E(X) = (4,100 \cdot 0.9512) + (-78,900 \cdot 0.0488) \]

Step 1: Calculate the first term

\[ 4,100 \cdot 0.9512 = 3,899.92 \]

Step 2: Calculate the second term

\[ -78,900 \cdot 0.0488 = -3,850.32 \]

Step 3: Combine the results

\[ E(X) = 3,899.92 + (-3,850.32) = 3,899.92 - 3,850.32 = 49.60 \]

Conclusion

The expected value of the profit \( E(X) \) for the insurance company, correctly calculated, is $49.60.

Thank you for your correction; if you have any further questions or need additional assistance, feel free to ask!