Original practice · fully worked
Original variant: survival through changing weekly risks
A field sensor is active at the end of week 2. Conditional on being active at the start of a week, its failure probabilities during weeks 3, 4, and 5 are 0.10, 0.15, and 0.20, respectively. Calculate the probability that the sensor remains active through the end of week 5.
- A
0.003 - B
0.388 - C
0.450 - D
0.550 - E
0.612
Answer and calculation
Convert each age-specific failure probability to a one-week conditional survival probability. Multiplying 0.90, 0.85, and 0.80 gives 0.612, so choice E.
Setup
Setup
The known survival through week two makes week three the first remaining interval. Complement each upcoming conditional failure probability.
q3=1−0.10=0.90 q4=1−0.15=0.85 q5=1−0.20=0.80 Model
Model
Remaining active through week five requires survival in all three successive weeks.
Pr(active through week 5∣active after week 2)=q3q4q5 Compute
Compute
Multiply the conditional survival probabilities in chronological order.
q3q4q5=(0.90)(0.85)(0.80)=0.612 Answer
Answer
The sensor has a 61.2% chance of staying active throughout the three remaining weeks.
0.612(E) 