This Exam P sample reference tests Inclusion-Exclusion. This problem first reconstructs overlapping purchase groups by inclusion-exclusion and then prices each disjoint group. The overlap contains 30 customers, and the resulting total revenue is 17,800, which selects choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AStarting from the no-discount marginal revenue of 19,000 and subtracting all 10,800 of discounted joint-customer revenue gives 8,200. Only the 1,200 discount, not the entire joint revenue, should be subtracted.
CThis equals 19,000-600 and therefore applies a 20-per-customer reduction to the 30 joint buyers. The required 10% reduction is 40 per joint buyer.
DAssuming the two purchases are independent gives 200(0.20)(0.35)=14 joint buyers. Subtracting their 40-per-customer discount from 19,000 gives 18,440, but independence was not given.
EThis is 19,000+10,800, which adds the discounted joint revenue after the joint buyers have already been included in both marginal revenue terms.
Original practice · fully worked
Original variant: revenue from overlapping software add-ons
Among 100 software orders, 45% include analytics, 30% include archival storage, and 51% include exactly one of those two add-ons. Analytics alone costs 20 dollars, storage alone costs 30 dollars, and an order containing both is charged a 40-dollar bundle price. Calculate the expected total add-on revenue from the 100 orders.
A 480
B 1,200
C 1,680
D 1,800
E 2,280
Variant answer in brief
If x is the probability of both add-ons, then 0.45+0.30-2x=0.51, so x=0.12. Pricing the analytics-only, storage-only, and bundle groups gives expected revenue 1,680, so choice C.
Setup
Setup
Let x be the probability that an order includes both add-ons. The exactly-one probability counts each marginal group after removing the overlap twice.
0.45+0.30−2x=0.51
Model
Model
Solve for the overlap and then obtain the two exclusive purchase probabilities.
x=0.12
Pr(A only)=0.45−0.12=0.33
Pr(S only)=0.30−0.12=0.18
Compute
Compute
Compute expected add-on revenue per order and scale to 100 orders.
E[R1]=20(0.33)+30(0.18)+40(0.12)=16.80
E[R100]=100(16.80)=1,680
Answer
Answer
The expected total add-on revenue is 1,680 dollars.
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