The expected value of an insurance policy helps the company determine whether the premium charged covers the potential payout from accidents.

To calculate the expected value of the insurance policy, we'll use the following formula:

$\text{Expected Value} = (\text{Probability of no accident} \times \text{Premium}) + (\text{Probability of accident} \times (\text{Premium} - \text{Payout}))$

Step 1: Define variables

• Probability of accident = $2.3\% = 0.023$
• Probability of no accident = $1 - 0.023 = 0.977$
• Payout = $3400$ dollars
• Premium = $190$ dollars

Step 2: Calculate the expected value

Case 1: No accident

The company keeps the full premium of $190 if no accident occurs. The probability of this is 0.977.

Expected value from no accident: $0.977 \times 190 = 185.63$

Case 2: Accident occurs

If an accident occurs, the company has to pay out $3400, so their net result is the premium minus the payout:

Net result from accident: $190 - 3400 = -3210$

The probability of an accident is 0.023. So, the expected value from this case is: $0.023 \times (-3210) = -73.83$

Step 3: Total expected value

Now, sum the expected values from both cases: $185.63 + (-73.83) = 111.80$

Thus, the expected value of this insurance policy for the company is $111.80.

This means, on average, the company expects to make a profit of $111.80 from this policy.

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Would you like more details on how these steps were calculated or need further clarification?

Here are five related questions to deepen your understanding:

1. What would happen to the expected value if the probability of an accident increased to 5%?
2. How does increasing the premium affect the company’s expected profit?
3. How would a higher average payout (e.g., $5000) influence the expected value?
4. If the customer’s risk were reduced to 1%, how would the expected value change?
5. What is the break-even premium for this policy?

Tip: Understanding expected value is key to assessing risks and profits in fields like insurance and gambling.