Financial Impact of Discount Vouchers

Solution & Explanation

To evaluate the financial impact of distributing $5 discount vouchers to N customers, we need to calculate the expected cost to the company. This problem can be approached using the concept of expected value in a binomial distribution, which is a common scenario in probability and statistics.

Understanding the Problem

• N: Number of customers receiving the voucher.
• P: Probability that any given customer will use the voucher.
• Voucher Value: $5.

The goal is to determine the expected financial burden or cost for the company when these vouchers are distributed.

Binomial Distribution

When we have a fixed number of independent trials (in this case, N customers), each with the same probability of success (where "success" is a customer using the voucher), the scenario follows a binomial distribution. In a binomial distribution:

• E[X] = N * P

Where E[X] is the expected number of successes (customers using the voucher).

Expected Financial Cost

The expected cost to the company is the expected number of customers who use the voucher multiplied by the value of each voucher:

• Expected Number of Voucher Uses: E[X] = N * P
• Expected Cost: Voucher Value * E[X] = $5 * (N * P)

Thus, the expected financial burden or cost to the company is:

$E[\text{Total Cost}] = 5 \times N \times P$

Explanation of the Calculation

1. Expected Number of Voucher Uses (N * P):

   • This figure represents the average number of customers who are expected to redeem the voucher. Since each customer independently decides whether to use the voucher, the binomial distribution provides a straightforward way to calculate this.

2. Expected Financial Cost (5 * N * P):

   • The expected cost is derived by multiplying the expected number of voucher uses by the value of each voucher ($5). This gives the total expected cost the company will incur.

Conclusion

The financial impact of distributing the vouchers is quantified by the expected cost, calculated as 5 times the number of customers (N) times the probability of voucher usage (P). This formula provides a simple yet powerful way to assess the potential cost implications of promotional strategies involving discount vouchers.