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?
To find the expected gain per package, we need to calculate the probability of the company failing to deliver within 24 hours and the amount they gain or lose in each scenario.
Let's start by calculating the probability of the company failing to deliver within 24 hours. We are given that this occurs only 3% of the time, so the probability is 0.03 or 3/100.
Now, let's consider the gain or loss in each scenario:
- If the company delivers within 24 hours, they gain $17.95 - $15.50 = $2.45 per package.
- If the company fails to deliver within 24 hours, they lose the full shipping charge of $17.95 per package.
To calculate the expected gain per package, we multiply the probability of each scenario by its respective gain or loss and sum them up:
Expected gain = (Probability of delivery within 24 hours * Gain per package) + (Probability of failure to deliver * Loss per package)
Expected gain = (0.97 * $2.45) + (0.03 * -$17.95)
Calculating this, we get:
Expected gain = $2.3815 + (-$0.5385) = $1.843
Therefore, the shipping company's expected gain per package is $1.843.