Question

From experience, a shipping company knows that the cost of delivering a small package within 24 hours is \(\$ 14.80 .\) The company charges \(\$ 15.50\) for shipment but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only \(2 \%\) of its packages within the 24 -hour period, what is the expected gain per package?

Short answer

Answer: The expected gain per package for the shipping company is $0.376.

Step-by-step solution

Identify variables and probabilities

In this problem, we need to identify some variables: - The cost of delivering a package within 24 hours: `$14.80` - The amount charged for shipment: `$15.50` - The probability of succeeding in delivering within 24 hours: `98%` - The probability of failing to deliver within 24 hours: `2%`

Calculate the expected value

To find the expected gain per package, we use the formula for expected value, which is given by: Expected Value = (Probability of success) x (Value if success) + (Probability of failure) x (Value if failure) Here, the probability of success refers to delivering the package within 24 hours, and the probability of failure refers to not delivering the package within 24 hours. The value if success is the difference between the amount charged and the cost (shipment charge - delivery cost), and the value if failure is the negative of the shipment charge (as the company will refund the charge).

Plug in the values

Now we can plug in the values to the formula: Expected Value = (0.98) x (\(15.50 - \)14.80) + (0.02) x (-$15.50)

Calculate the result

Performing the calculations gives us: Expected Value = (0.98) x (\(0.70) + (0.02) x (-\)15.50) Expected Value = \(0.686 - \)0.31 Expected Value = $0.376 The shipping company has an expected gain of `$0.376` per package.

Key concepts

Probability of Success

In the context of the shipping company exercise, the "Probability of Success" is essentially the chance that a package is delivered within 24 hours as promised. This probability is given as 98%, or in other words, the company succeeds in making timely deliveries 98 times out of every 100 shipments.

The probability of success can be written in decimal form for calculations as 0.98. This is a crucial figure in calculating expected gains because it accounts for the vast majority of successful transactions where the company does not have to issue a refund.

Understanding this concept involves recognizing that a higher probability of success typically means the company is efficient and reliable. As a result, they can expect more profit per package as fewer refunds are needed. Success probabilities are often determined through historical data or past experiences, providing a realistic expectation for future outcomes.

Probability of Failure

The "Probability of Failure" complements the probability of success. It represents the likelihood that the shipping company does not manage to deliver a package within 24 hours, thus needing to refund the delivery charge to the customer. In this exercise, the probability of failure is stated to be 2% or 0.02 in decimal form.

While this percentage is low, it plays a critical role in determining the overall expected gain for the company. A minor proportion of faulty deliveries can still affect the expected value, hence businesses strive to minimize this probability as much as possible.

A deeper understanding of this concept involves considering measures a company can take to reduce the probability of failure, such as optimizing routes, increasing workforce efficiency, or adopting advanced technology for better tracking. Each reduction in the probability of failure directly impacts profitability by lowering refund instances.

Expected Gain

Expected gain refers to the profit a company can anticipate per transaction after accounting for both successes and failures. To find the expected gain, the company uses a specific formula which incorporates probability.For this exercise, the expected gain per package is calculated using the equation:\[Expected \ Value = (Probability \ of \ success) \times (Value \ if \ success) + (Probability \ of \ failure) \times (Value \ if \ failure)\]Here, "Value if success" is the profit made when a delivery is successful, calculated by subtracting the delivery cost from the charged amount: \(15.50 - 14.80 = 0.70\). "Value if failure" considers that the entire charge is refunded, leading to a negative value, \(-15.50\). Substituting these values, the expected value is:\( Expected \ Value = (0.98) \times (0.70) + (0.02) \times (-15.50)\)\( Expected \ Value = 0.686 - 0.31\)\( Expected \ Value = 0.376\)Thus, the shipping company can expect a gain of $0.376 per package. This calculation helps companies make informed decisions about pricing and operational strategies.

Cost and Revenue Analysis

Cost and revenue analysis involves evaluating the expenses incurred against the money earned to determine profitability. In this scenario, the shipping company charges $15.50 per package and incurs a cost of $14.80 for delivery. The difference between these amounts, $0.70, represents the gross profit when a delivery is successful.

However, for complete analysis, the company must also factor in the refunds issued in cases of delivery failures. This refund is $15.50 per delayed package, significantly affecting the overall profitability due to the probability of failure.

Analyzing costs involves:

• Understanding fixed and variable costs. Here, it is largely variable as it directly correlates with package deliveries.
• Implementing strategies to minimize costs without compromising service quality, such as optimizing delivery routes or negotiating better rates.

Revenue analysis involves:

• Forecasting sales based on historical performance data.
• Setting competitive shipping rates to attract customers while ensuring profitability.

In conclusion, through thoughtful cost and revenue analysis, companies can effectively plan to enhance their financial performance and make strategic improvements.