Rideshare Voucher Expected Cost
A rideshare company gives N riders a $5 voucher each. Each rider uses the voucher with an independent probability p. What is the expected total cost to the company for these vouchers?
Related Concepts
Hint
- Focus on one rider first:
- What's the cost to the company if this one rider uses the voucher? What's the probability?
- What's the cost if this one rider does NOT use the voucher? What's the probability?
- Calculate the expected cost for this single rider.
- Total Expected Cost:
- How does the expected cost for one rider relate to the total expected cost for N riders? (Think about the Linearity of Expectation).
Explanation: Rideshare Voucher Expected Cost
Imagine you're giving out $5 coupons to 100 people. Each person might use it or might not. Let's say there's a 30% chance (probability p=0.3) any given person will use their coupon. How much money do you expect this promotion to cost you in total?
- Cost for ONE person:
- If they use it (30% chance): Cost is $5.
- If they don't use it (70% chance): Cost is $0.
- Cost for ALL 100 people: Since each person's decision is separate, the total expected cost is just the expected cost for one person multiplied by the number of people.
Total Expected Cost = 100 people × $1.50/person = $150.
So, even though each coupon is $5, because not everyone uses it, your average expected cost is lower.
We need to find the expected total cost to the company. We can do this by first finding the expected cost for a single rider, and then using the linearity of expectation to find the total expected cost for N riders.
1. Expected Cost for a Single Rider
Let Ci be the cost associated with rider i.
The value of the voucher is $5.
The probability that a rider uses the voucher is p.
The probability that a rider does not use the voucher is (1-p).
For a single rider, the possible costs to the company are:
- Cost = $5, if the rider uses the voucher (with probability p).
- Cost = $0, if the rider does not use the voucher (with probability 1-p).
The expected cost for one rider, E[Ci], is calculated as:
E[Ci] = (Cost if used × P(uses)) + (Cost if not used × P(not used))
E[Ci] = ($5 × p) + ($0 × (1-p))
E[Ci] = $5p + $0
E[Ci] = $5p
So, the expected cost for each individual rider is $5p.
2. Total Expected Cost for N Riders
Let Ctotal be the total cost for N riders.
Ctotal = C₁ + C₂ + ... + CN (the sum of costs for each rider).
By the linearity of expectation, the expected value of a sum of random variables is the sum of their expected values. This holds true even if the variables are dependent, but in this case, each rider's decision to use the voucher is stated as independent.
E[Ctotal] = E[C₁ + C₂ + ... + CN]
E[Ctotal] = E[C₁] + E[C₂] + ... + E[CN]
Since the expected cost for each rider is the same ($5p):
E[Ctotal] = $5p + $5p + ... + $5p (N times)
E[Ctotal] = N × ($5p)
E[Total Cost] = $5Np
Example Calculation
If N = 100 riders and the probability of a rider using the voucher is p = 0.3 (30%):
Expected cost per rider = $5 × 0.3 = $1.50
Total Expected Cost = 100 × $1.50 = $150
Alternatively, using the formula directly:
Total Expected Cost = $5 × 100 × 0.3 = $150
Linearity of Expectation is Key: The crucial property used here is that E[X+Y] = E[X] + E[Y]. This allows us to simply sum the individual expected costs without needing to consider the complex probability distribution of the total number of riders who might use the voucher (which would be a Binomial distribution).
Business Decision: If the company expects to make an average profit of $2 per rider after accounting for voucher costs, what would be the maximum probability 'p' they could tolerate for voucher usage if N=1000 riders?