Questions: Avicenna, an insurance company, offers five-year commercial property insurance policies to small businesses. If the holder of one of these policies experiences property damage in the next five years, the company must pay out 26,500 to the policy holder. Executives at Avicenna are considering offering these policies for 497 each. Suppose that for each holder of a policy there is a 2% chance they will experience property damage in the next five years and a 98% chance they will not. (If necessary, consult a list of formulas.) If the executives at Avicenna know that they will sell many of these policies, should they expect to make or lose money from offering them? How much? To answer, take into account the price of the policy and the expected value of the amount paid out to the holder. Avicenna can expect to make money from offering these policies. In the long run, they should expect to make dollars on each policy sold. Avicenna can expect to lose money from offering these policies. In the long run, they should expect to lose dollars on each policy sold. Avicenna should expect to neither make nor lose money from offering these policies.

Avicenna, an insurance company, offers five-year commercial property insurance policies to small businesses. If the holder of one of these policies experiences property damage in the next five years, the company must pay out 26,500 to the policy holder. Executives at Avicenna are considering offering these policies for 497 each. Suppose that for each holder of a policy there is a 2% chance they will experience property damage in the next five years and a 98% chance they will not. (If necessary, consult a list of formulas.) If the executives at Avicenna know that they will sell many of these policies, should they expect to make or lose money from offering them? How much? To answer, take into account the price of the policy and the expected value of the amount paid out to the holder. Avicenna can expect to make money from offering these policies. In the long run, they should expect to make dollars on each policy sold. Avicenna can expect to lose money from offering these policies. In the long run, they should expect to lose dollars on each policy sold. Avicenna should expect to neither make nor lose money from offering these policies.
Transcript text: Avicenna, an insurance company, offers five-year commercial property insurance policies to small businesses. If the holder of one of these policies experiences property damage in the next five years, the company must pay out $\$ 26,500$ to the policy holder. Executives at Avicenna are considering offering these policies for $\$ 497$ each. Suppose that for each holder of a policy there is a $2 \%$ chance they will experience property damage in the next five years and a $98 \%$ chance they will not. (If necessary, consult a list of formulas.) If the executives at Avicenna know that they will sell many of these policies, should they expect to make or lose money from offering them? How much? To answer, take into account the price of the policy and the expected value of the amount paid out to the holder. Avicenna can expect to make money from offering these policies. In the long run, they should expect to make dollars on each policy sold. Avicenna can expect to lose money from offering these policies. In the long run, they should expect to lose $\square$ dollars on each policy sold. Avicenna should expect to neither make nor lose money from offering these policies.
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Solution

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Solution Steps

To determine whether Avicenna will make or lose money from offering these insurance policies, we need to calculate the expected value of the payout and compare it to the price of the policy. The expected value (EV) of the payout is calculated by multiplying the probability of property damage by the payout amount and adding it to the probability of no damage multiplied by zero (since no payout is made in that case). Then, subtract the expected payout from the policy price to determine the expected profit or loss per policy.

Step 1: Calculate the Expected Payout

To find the expected payout, we use the formula for expected value: \[ \text{Expected Payout} = (\text{Probability of Damage} \times \text{Payout}) + (\text{Probability of No Damage} \times 0) \] Substituting the given values: \[ \text{Expected Payout} = (0.02 \times 26500) + (0.98 \times 0) = 530 \]

Step 2: Calculate the Expected Profit or Loss per Policy

The expected profit or loss per policy is calculated by subtracting the expected payout from the policy price: \[ \text{Expected Profit per Policy} = \text{Policy Price} - \text{Expected Payout} \] Substituting the given values: \[ \text{Expected Profit per Policy} = 497 - 530 = -33 \]

Step 3: Determine if Avicenna Will Make or Lose Money

Since the expected profit per policy is negative, Avicenna can expect to lose money on each policy sold. Specifically, they will lose \(-33\) dollars per policy.

Final Answer

Avicenna can expect to lose money from offering these policies. In the long run, they should expect to lose \(\boxed{33}\) dollars on each policy sold.

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