Let's break down each part of the problem. For each part, we’ll calculate the total amount the person pays with and without insurance. When calculating with insurance, we’ll add the deductible for each claim to the total premiums paid over the two-year period. When calculating without insurance, the person would simply pay the full amount of the claims.

Part (a)

• Deductible: $800 per claim
• Claims: $510 and $950
• Annual Premium: $580
• Period: 2 years

With Insurance:

• Total premiums for 2 years = $580 \times 2 = 1160$
• For the $510 claim, the person pays the full amount since it is less than the deductible (i.e., $510).
• For the $950 claim, the person pays the deductible ($800), and the insurance covers the rest ($950 - $800 = $150, paid by the insurance).

Total paid with insurance: $\text{Total paid} = 1160 \ (\text{premium}) + 510 \ (\text{claim}) + 800 \ (\text{deductible}) = 2470$

Without Insurance:

• The person pays the full amount of both claims: $510 + 950 = 1460$.

$\text{Without insurance} = 1460$

Part (a) Answers:

• With insurance: $2470
• Without insurance: $1460

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Part (b)

• Deductible: $200 per claim
• Claims: $510 and $1350
• Annual Premium: $820
• Period: 2 years

With Insurance:

• Total premiums for 2 years = $820 \times 2 = 1640$
• For the $510 claim, the person pays the deductible ($200), and insurance covers $510 - 200 = 310$.
• For the $1350 claim, the person pays the deductible ($200), and insurance covers $1350 - 200 = 1150$.

Total paid with insurance: $\text{Total paid} = 1640 \ (\text{premium}) + 200 \ (\text{deductible}) + 200 \ (\text{deductible}) = 2040$

Without Insurance:

• The person pays the full amount of both claims: $510 + 1350 = 1860$.

$\text{Without insurance} = 1860$

Part (b) Answers:

• With insurance: $2040
• Without insurance: $1860

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Part (c)

• Deductible: $1000 per claim
• Claims: $220 and $1700
• Annual Premium: $380
• Period: 2 years

With Insurance:

• Total premiums for 2 years = $380 \times 2 = 760$
• For the $220 claim, the person pays the full amount since it is less than the deductible ($220).
• For the $1700 claim, the person pays the deductible ($1000), and insurance covers $1700 - 1000 = 700$.

Total paid with insurance: $\text{Total paid} = 760 \ (\text{premium}) + 220 \ (\text{claim}) + 1000 \ (\text{deductible}) = 1980$

Without Insurance:

• The person pays the full amount of both claims: $220 + 1700 = 1920$.

$\text{Without insurance} = 1920$

Part (c) Answers:

• With insurance: $1980
• Without insurance: $1920

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Summary of Answers:

• Part (a):
  • With insurance: $2470
  • Without insurance: $1460
• Part (b):
  • With insurance: $2040
  • Without insurance: $1860
• Part (c):
  • With insurance: $1980
  • Without insurance: $1920

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Would you like more details or clarifications on any part?

Here are five related questions to consider:

1. How does increasing the deductible impact the total cost with insurance?
2. What would happen if there were more frequent claims?
3. How does the annual premium influence whether a person benefits from having insurance?
4. Is there a deductible level that might make the insurance not worth it?
5. How do different insurance policies (like health or home insurance) apply similar deductible principles?

Tip: Always consider both the deductible and premium when evaluating the overall value of an insurance policy.