Original practice · fully worked
Original variant: paired exponential component lifetimes
Two independent components have identical exponential lifetimes. The probability that at least one component is still operating after three hours is 0.75. Calculate the probability that both components are still operating after six hours.
- A
0.1250 - B
0.2500 - C
0.5000 - D
0.0625 - E
0.7500
Answer and calculation
The observation makes each component's three-hour survival probability 0.5. Exponential scaling gives six-hour survival 0.25 per component, and independence gives 0.25²=0.0625, so choice D is correct.
Setup
Setup
Let S(t) be the common component survival function. Complement the observed at-least-one-survivor event.
Pr(both fail by 3)=1−0.75=0.25 Model
Model
Independence makes the probability that both have failed by time three the square of the individual failure CDF.
(1−S(3))2=0.25 1−S(3)=0.5⟹S(3)=0.5 Compute
Compute
Use exponential survival over two consecutive three-hour intervals, then require both independent components to survive.
S(6)=S(3)2=(0.5)2=0.25 Pr(both survive to 6)=S(6)2=(0.25)2=0.0625 Answer
Answer
The probability that both components remain operating after six hours is 0.0625.
0.0625(D) 