This Exam P sample reference tests Binomial Distribution. One group has at least nine completers with probability 0.37581. Exactly one of two independent groups does so with probability 0.46915, choice E.
For one group, at least nine completers means exactly nine or exactly ten completers in a binomial count.
q=(910)(0.8)9(0.2)+(0.8)10=0.375810
Model
Model
The two groups are independent. Exactly one group meets the threshold through either of two disjoint orders: the first succeeds and second fails, or vice versa.
Pr(exactly one group)=q(1−q)+(1−q)q
Compute
Compute
The one-group probability is approximately 0.375810. Doubling the product of this probability and its complement gives approximately 0.469154.
2q(1−q)=0.469154
Answer
Answer
The probability that exactly one group meets the completion threshold is approximately 0.469, selecting choice E.
0.469(E)
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CThe value 0.235 is half the correct result. It counts only one order for which group meets the threshold.
DThe value 0.376 is the probability that one specified group has at least nine completers, not the probability that exactly one of two groups does.
Original practice · fully worked
Original variant: complete participation in exactly two teams
Three independent teams each consist of eight members. Every member independently finishes a task with probability 0.90. Find the probability that all eight members finish in exactly two of the three teams.
A 0.18530
B 0.31661
C 0.43047
D 0.49033
E 0.56953
Variant answer in brief
A team is fully complete with probability q=0.9⁸. Selecting the two complete teams gives 3q²(1-q)=0.31661, choice B.
Setup
Setup
A team is fully complete only when all eight independent members finish, giving one-team completion probability approximately 0.430467.
q=(0.90)8=0.430467
Model
Model
Across three independent teams, the number of fully complete teams is binomial. Exactly two complete teams can be chosen in three ways.
Pr(exactly two full teams)=(23)q2(1−q)
Compute
Compute
Evaluating the exactly-two binomial mass gives approximately 0.316607.
3q2(1−q)=0.316607
Answer
Answer
The probability that exactly two teams are fully complete is approximately 0.31661, selecting choice B.
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.
