Coupon Distribution Analysis

Coupon Distribution Analysis

Problem Overview

In a ride-sharing application, we need to determine the budget for a coupon distribution initiative and calculate certain probabilities related to the distribution of coupons among riders.

Solution & Explanation

1. Budget Calculation

   • Modeling the Distribution:
     • The number of coupons distributed can be modeled using a Binomial Distribution.
     • Parameters:
       • N: Total number of riders
       • p: Probability of a rider receiving a coupon
       • C: Number of coupons distributed, where $C \sim B(N, p)$
   • Expected Budget:
     • The expected number of coupons distributed is $E[C] = N \times p$.
     • The expected cost for distributing these coupons is $5 \times E[C] = 5 \times N \times p$.
   • Standard Deviation:
     • $\text{Std}[C] = \sqrt{N \times p \times (1-p)}$
   • Budget with Safety Margin:
     • To account for variability, add one standard deviation to the expected cost:
     • Total budget $B = 5 \times (N \times p + \sqrt{N \times p \times (1-p)})$

2. Probability Calculations

   • Both Riders Receiving a Coupon:
     • Let $C_1$ and $C_2$ be the events where the first and second rider receive the coupon.
     • Assuming independence, the probability both receive a coupon is:
       • $P(C_1 \cap C_2) = p^2$
   • Only One Rider Receiving a Coupon:
     • The probability that exactly one of the two riders receives a coupon:
     • This can happen in two ways:
       • First rider receives, second does not: $P(C_1 \cap C_2^c) = p \times (1-p)$
       • Second rider receives, first does not: $P(C_1^c \cap C_2) = (1-p) \times p$
     • Total probability for only one receiving a coupon:
       • $P(C_1 \cap C_2^c) + P(C_1^c \cap C_2) = 2p(1-p)$

Conclusion

The budget for the coupon initiative is calculated by considering both the expected number of coupons distributed and the variability in distribution. Meanwhile, the probability calculations help understand the likelihood of one or both riders receiving a coupon, providing insights into the coupon distribution dynamics among riders.