This Exam P sample reference tests Conditional Probability. The relevant nested-event slice has probability 0.48-0.36=0.12, while the conditioning event has probability 1-0.36=0.64. Their ratio is 0.1875, which rounds to 0.19 and selects choice B.
How to solve this Conditional Probability question
Setup
Setup
Let A_2 be the event of reaching and passing the second stage, and let A_3 be the event of completing all three stages successfully. The process makes A_3 a subset of A_2.
A3⊂A2
Pr(A2)=0.48,Pr(A3)=0.36
Model
Model
Within the condition that full completion does not occur, passing the second stage means belonging to A_2 but not A_3.
Pr(A2∣A3c)=Pr(A3c)Pr(A2∩A3c)
Pr(A2∩A3c)=Pr(A2)−Pr(A3)
Compute
Compute
Evaluate the numerator as the difference of nested probabilities and the denominator as a complement.
Pr(A2∩A3c)=0.48−0.36=0.12
Pr(A3c)=1−0.36=0.64
Pr(A2∣A3c)=0.640.12=0.1875
Answer
Answer
The requested conditional probability rounds to 0.19.
0.19(B)
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AThis is the joint numerator 0.48-0.36=0.12. It has not been divided by the conditioning probability 0.64.
CThis divides the same numerator by 0.48, giving 0.12/0.48=0.25. That conditions on passing the second stage instead of on not completing all three.
DThis is only the probability of the conditioning event, 1-0.36=0.64, rather than the conditional ratio.
EThis computes 0.36/0.48=0.75, the conditional probability of completing the third stage among those who passed the second.
Original practice · fully worked
Original variant: condition on holding exactly one credential
Among a group of analysts, 55% hold credential A, 40% hold credential B, and 25% hold neither credential. An analyst is selected from those who hold exactly one of the two credentials. Calculate the probability that the selected analyst holds B but not A.
A 0.2000
B 0.2500
C 0.3636
D 0.4000
E 0.5500
Variant answer in brief
The union probability is 0.75, so the overlap is 0.20. This leaves 0.20 in B only and 0.55 in exactly one credential, giving 0.20/0.55=0.3636 and choice C.
Setup
Setup
Let A and B denote the two credential events. First convert the neither probability into the union probability.
Pr(A∪B)=1−0.25=0.75
Model
Model
Use inclusion-exclusion to recover the overlap, then split the union into its A-only, B-only, and both parts.
Pr(A∩B)=0.55+0.40−0.75=0.20
Pr(B∩Ac)=0.40−0.20=0.20
Compute
Compute
The exactly-one event consists of the two nonoverlapping one-credential regions.
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