This Exam P sample reference tests Conditional Hypergeometric Counting. Given six sales and equal independent purchase chances, every six-person customer subset is equally likely. Selecting two from each interest group gives 90/924=0.0974, choice D.
How to solve this Conditional Hypergeometric Counting question
Setup
Setup
Conditional on exactly six sales, all six-person subsets of the 12 customers are equally likely because the individual purchase probabilities are the same.
Nall=(612)
Model
Model
A favorable subset chooses two customers from each of the three interest groups. Multiply the three combination counts.
Nfav=(26)(24)(22)=90
Compute
Compute
There are 90 favorable subsets among 924 total six-person subsets, giving approximately 0.097403.
Pr=(612)90=15415=0.097403
Answer
Answer
The conditional composition probability is about 0.097.
0.097(D)
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Original variant: balanced species in a selected team
A shelter has 5 cats, 4 dogs, and 3 rabbits. Six animals are selected uniformly without replacement. Find the probability that the selected group contains exactly two of each species.
A 0.0974
B 0.1299
C 0.1623
D 0.1948
E 0.2597
Variant answer in brief
The favorable groups number C(5,2)C(4,2)C(3,2)=180 among C(12,6)=924 total groups, giving 0.1948, choice D.
Setup
Setup
Every six-animal subset is equally likely under uniform sampling without replacement.
Nall=(612)=924
Model
Model
A balanced subset chooses two of the five cats, two of the four dogs, and two of the three rabbits.
Nfav=(25)(24)(23)=180
Compute
Compute
The favorable count is 180 and the total count is 924, so the probability is approximately 0.194805.
Pr=924180=0.194805
Answer
Answer
The balanced-selection probability is about 0.1948.
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