This Exam P sample reference tests Conditional Probability. The three-way cell is 0.06. The none cell is 0.28 and the not-A total is 0.60, yielding 0.28/0.60=0.4667, choice C.
How to solve this Conditional Probability question
Setup
Setup
Let x be the probability of the three-way cell. Solve the supplied conditional ratio to obtain x=0.06.
x+0.12x=31,x=0.06
Model
Model
The eight membership cells partition the sample space. Subtract the three one-event cells, three two-event cells, and the three-way cell to obtain the none cell.
p0=1−3(0.10)−3(0.12)−0.06=0.28
Compute
Compute
The none cell is 0.28 and the conditioning event A complement has probability 0.60, so the conditional ratio is 0.28/0.60=0.4667.
Pr(Ac)=1−0.10−2(0.12)−0.06=0.60
Pr(none∣Ac)=0.28/0.60
Answer
Answer
The requested conditional probability rounds to 0.467, corresponding to choice C.
0.467(C)
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AThe value 0.280 is the unconditional probability of the none cell. Conditioning on A complement requires division by 0.60.
BThe value 0.311 is 0.28/0.90 to rounding, using an incorrect conditioning denominator. The event A complement has probability 0.60.
Original practice · fully worked
Original variant: unmarked share from a count-of-tags distribution
A labeling system assigns zero to three tags to each record. The probabilities of exactly one tag, exactly two tags, and exactly three tags are 0.36, 0.24, and 0.08. Determine the probability that a record receives no tag.
A 0.08
B 0.24
C 0.28
D 0.32
E 0.68
Variant answer in brief
The four tag-count outcomes partition the sample space, so the missing mass is 1-0.68=0.32, choice D.
Setup
Setup
Let N be the number of tags. The four possible counts 0, 1, 2, and 3 exhaust the sample space.
k=0∑3Pr(N=k)=1
Model
Model
Use normalization of the probability mass function and solve for the only unreported mass, Pr(N=0).
Pr(N=0)=1−Pr(N=1)−Pr(N=2)−Pr(N=3)
Compute
Compute
The positive tag-count probabilities sum to 0.36+0.24+0.08=0.68, leaving probability 0.32 for no tag.
Pr(N=0)=1−0.36−0.24−0.08=0.32
Answer
Answer
Therefore a record receives no tag with probability 0.32, which is choice D.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
