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Let's solve the questions based on the provided image.

### Part (a)
**What is the probability that Hana will die in her 60th year?**

The probability of death in her 60th year is given as $ 0.00574 $.

Next, the expected cost to the insurance company is calculated using the formula:
\[
\text{Expected cost} = P(\text{death}) \times \text{benefit}
\]
For the 60th year:
\[
\text{Expected cost} = 0.00574 \times 50,000 = 287
\]

### Part (b)
We repeat the same calculations for ages 61, 62, 63, and 64.

#### Age 61
The probability of death is $ 0.00854 $.
\[
\text{Expected cost} = 0.00854 \times 50,000 = 427
\]

#### Age 62
The probability of death is $ 0.00890 $.
\[
\text{Expected cost} = 0.00890 \times 50,000 = 445
\]

#### Age 63
The probability of death is $ 0.01056 $.
\[
\text{Expected cost} = 0.01056 \times 50,000 = 528
\]

#### Age 64
The probability of death is $ 0.01186 $.
\[
\text{Expected cost} = 0.01186 \times 50,000 = 593
\]

**Total Expected Cost** over the years 60–64 is the sum of the individual expected costs:
\[
287 + 427 + 445 + 528 + 593 = 2,280
\]

### Part (c)
**If the insurance company wants to make a profit of $700 above the expected total cost, how much should it charge?**

To calculate the charge for the policy:
\[
\text{Charge} = \text{Total expected cost} + \text{Profit}
\]
\[
\text{Charge} = 2,280 + 700 = 2,980
\]

### Part (d)
**If the insurance company charges $5,000 for the policy, how much profit does the company expect to make?**

The profit is calculated as:
\[
\text{Profit} = \text{Charge} - \text{Total expected cost}
\]
\[
\text{Profit} = 5,000 - 2,280 = 2,720
\]

---

Do you have any questions or want more details on any step? Here are 5 related questions:

1. How does the expected cost calculation change if the death benefit increases?
2. What would happen to the insurance premium if the probability of death significantly changes at one age?
3. How is the expected cost affected by a higher life expectancy (beyond 65)?
4. How would the insurance premium change if the insurance policy also covered accidental death with a different benefit amount?
5. What would the total expected cost be if the probabilities for each age doubled?

### Tip: 
When calculating expected values in insurance problems, always use the actual probability for each year to avoid under- or overestimating the cost.

Hana is a 60-year-old Asian female in reasonably good health. She wants to take out a $50,000 term life insurance policy until she is 65. The policy will expire on her 65th birthday. If she dies before her 65th birthday, the insurance company will pay out $50,000 to her beneficiaries. The problem asks for calculations of the expected costs, total expected cost, policy charge with profit, and expected profit given specific probabilities of death in years 60 to 64.

Solve an insurance math problem involving expected value calculations. Learn how to compute expected costs and total expected costs over multiple years using probabilities. This detailed example covers the calculation of policy charge with profit and expected profit. Suitable for students in Grades 10-12.

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