From previous records a shipping company knows that the cost to deliver a small package within 24 hours is $15.50. The company charges $17.95 for shipping, but guarantees to refund the full charge if the delivery is not made within 24 hours. If the company fails to deliver within the 24 hour window only 3% of the time, what is their expected gain per package?

Well, I must say, this shipping company has quite a tricky pricing strategy. Let's do some funny math to figure out their expected gain per package!

First, let's find out the probability that the package is not delivered within 24 hours. We know that it only happens 3% of the time, so we can subtract that from 100% to get 97%.

Now, let's calculate their expected gain per package. The company charges $17.95 but only spends $15.50 to deliver it on time. So their actual gain per package, with on-time delivery, is $17.95 - $15.50 = $2.45.

Since the delivery is not made within 24 hours only 3% of the time, the company has to refund the full charge, which means they lose $17.95 for that 3% of the packages.

So, let's calculate the expected gain per package now. We multiply the gain of $2.45 by the probability of on-time delivery, which is 97%, and add the loss of $17.95 multiplied by the probability of late delivery, which is 3%.

Expected gain per package = ($2.45 * 0.97) + (-$17.95 * 0.03)
Expected gain per package = $2.3815 - $0.5385
Expected gain per package ≈ $1.84

So, based on my humorous calculations, it looks like their expected gain per package is approximately $1.84. Keep in mind that this is just an estimate and not a guarantee, just like their 24-hour delivery!

To calculate the expected gain per package for the shipping company, we need to consider two scenarios: when the delivery is made within 24 hours and when it is not.

Scenario 1: Delivery is made within 24 hours
In this case, the company charges $17.95 for shipping but incurs a cost of $15.50 to deliver the package. Therefore, the gain per package is $17.95 - $15.50 = $2.45.

Scenario 2: Delivery is not made within 24 hours
According to the information given, the company fails to deliver within 24 hours only 3% of the time. This means that 97% of the time the company does deliver within the promised timeframe. Since the company guarantees to refund the full charge in case of a late delivery, the gain per package is -$17.95 (negative value) in this scenario.

To calculate the expected gain per package, we need to average the gains from both scenarios, weighted by their respective probabilities.

Expected Gain = (Probability of Scenario 1 * Gain from Scenario 1) + (Probability of Scenario 2 * Gain from Scenario 2)
Expected Gain = (0.97 * $2.45) + (0.03 * -$17.95)

Calculating the equation:
Expected Gain = ($2.3765) - ($0.5385)
Expected Gain = $1.838

Therefore, the shipping company's expected gain per package is $1.838.