Original practice · fully worked
Original variant: navigation-beacon shock test
Eight sealed navigation beacons ride through the same shock test. Five use a standard casing and three use a ceramic casing. Each beacon independently fails with probability 0.20. After learning that exactly two beacons failed, calculate the probability that both failed beacons used the standard casing.
- A
0.1049 - B
0.2936 - C
0.3571 - D
0.3906 - E
0.6250
Answer and calculation
Given two failures among equally exposed beacons, every pair is equally likely. Ten of the 28 pairs come from the five standard-casing beacons, so the conditional probability is 5/14, approximately 0.3571, and choice C is correct.
Setup
Setup
Condition on the observed total of two failed beacons. Equal independent failure probabilities make the failed pair uniform among all pairs of the eight labeled beacons.
#{possible failed pairs}=(28)=28 Model
Model
A favorable pair contains two of the five standard-casing beacons.
#{favorable pairs}=(25)=10 Compute
Compute
The conditional probability is the favorable-pair count divided by the total pair count.
Pr(both standard∣two failures)=2810 2810=145=0.3571428571 Answer
Answer
Rounded to four decimal places, the result is 0.3571.
0.3571(C) 