Original practice · fully worked
Original variant: conditional warranty margin after one component survives
A paired-device warranty collects 60 dollars total and pays 300 dollars if either device fails during a year. The probabilities that both survive, only device A survives, and only device B survives are 0.90, 0.04, and 0.03. Given that A survives, find expected premium minus payout.
- A
27.45 dollars - C
40.65 dollars - B
47.23 dollars - D
50.32 dollars - E
60.00 dollars
Answer and calculation
Given A survives, B fails with probability 0.04/0.94. The expected payout is 12.766, leaving a margin of 47.234 dollars.
Setup
Setup
Given that device A survives, the admissible states are both devices surviving and only A surviving. Their total probability is 0.94.
P(A survives)=0.90+0.04=0.94 Model
Model
A payout occurs within this conditioning event only when B fails, whose joint-state probability is 0.04.
P(B fails∣A survives)=0.940.04 Compute
Compute
The conditional expected payout is approximately 12.77 dollars. Subtracting it from the 60-dollar premium gives approximately 47.23 dollars.
E[C∣A]=3000.940.04=12.7660 60−12.7660=47.2340 Answer
Answer
The conditional expected margin is approximately 47.23 dollars, selecting choice B.
47.23 dollars(B) 