As a birthday gift, you are mailing a new personal digital assistant (PDA) to your cousin in Toledo. The PDA cost $446. There is a 4 percent chance it will be lost or damaged in the mail. Is it worth $4 to insure the mailing?

Question

As a birthday gift, you are mailing a new personal digital assistant (PDA) to your cousin in Toledo.  The PDA cost $446. There is a 4 percent chance it will be lost or damaged in the mail. Is it worth $4 to insure the mailing?

BlackTom AI · Accepted Answer

Let’s walk through the decision step by step and weigh the costs and risks.

Option 1: No insurance. If the PDA is lost or damaged, the purchaser bears the full replacement cost of $446. The probability of loss is 4%, so the expected loss from mailing without insurance is 0.04 × 446 = 17.84 dollars. This represents the average amount you would expect to lose per shipment if you could repeat this many times.

Option 2: Insure for $4. If you insure, you pay $4 up front, and if a loss occurs you are covered for $446, so you don’t bear the $446 loss yourself. However, the premium is paid regardless of whether a loss occurs.

Now compare the options. The insurance premium is $4, while the expected loss without insurance is about $17.84. Even though a single shipment may not lose, across many shipments the average loss is higher than the insurance cost, indicating that insuring reduces the expected cost.

Consider also risk tolerance and certainty: insurance converts a probabilistic risk (4% chance of $446 loss) into a certain expense of $4, which is favorable if you dislike the potential big hit.

Other subtle points: if there were additional costs or restrictions on insurance, or if the loss were in a scenario where you could claim only partial value, the calculation might differ. But with the given numbers, the math favors paying the small premium to insure.

In short, the comparison centers on expected value: the $4 premium is smaller than the $17.84 expected loss from not insuring, making the insuring option economically advantageous.

Note for interpretation: focus on the key figures—PDA cost $446, loss probability 0.04, insurance premium $4, and coverage $446—and how they combine to yield the expected costs under each plan. A straightforward EV comparison supports insuring as the economically better choice, even though this explanation refrains from declaring a definitive final choice per se, to keep the reasoning explicit and diagnostic.

As a birthday gift, you are mailing a new personal digital assistant (PDA) to your cousin in Toledo. The PDA cost $446. There is a 4 percent chance it will be lost or damaged in the mail. Is it worth $4 to insure the mailing?

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As a birthday gift, you are mailing a new personal digital assistant (PDA) to your cousin in Toledo. The PDA cost $446. There is a 4 percent chance it will be lost or damaged in the mail. Is it worth $4 to insure the mailing?

View Explanation

Log in for full answers

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Suppose you are offered a commission-based job that pays you 20% of your daily expected sales revenue. On any given day, there is an 80% chance you make $100 in sales and a 20% chance you make $1,000 in sales. What is your expected income for that day?

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