Original practice · fully worked
Original variant: expected reserve after a multi-failure penalty
A server operator starts a week with a 600-dollar maintenance reserve. Five independent backup disks each fail with probability 0.10. If at least two disks fail, a one-time 150-dollar emergency charge is paid. Find the expected reserve remaining.
- A
565.00 dollars - B
575.78 dollars - C
581.46 dollars - E
587.78 dollars - D
592.50 dollars
Answer and calculation
The chance of at least two failures is 1-0.9⁵-5(0.1)0.9⁴=0.08146. The expected emergency charge is 12.219 dollars, leaving 587.781.
Setup
Setup
Let N be the number of failures among five independent disks. The reserve loses 150 dollars exactly when N is at least 2.
N∼Bin(5,0.1) R=600−150I{N≥2} Model
Model
Find the binomial upper tail by subtracting the zero- and one-failure probabilities from one.
P(N≥2)=1−0.95−5(0.1)(0.9)4=0.08146 Compute
Compute
The emergency-charge probability is 0.08146, so its expected cost is 12.219 dollars and the expected remaining reserve is 587.781.
E[R]=600−150(0.08146)=587.781 Answer
Answer
Expected remaining reserve is 587.78 dollars, corresponding to choice E.
587.78 dollars(E) 