This Exam P sample reference tests Hypergeometric Sampling. The given probability ratio forces eight of the 27 items to be insured. The hypergeometric probability of selecting exactly two insured items is 266/975=0.273, choice C.
How to solve this Hypergeometric Sampling question
Setup
Setup
Let the unknown insured count be determined from the ratio of the probabilities of selecting exactly one insured item and selecting none. Taking this ratio cancels the common sample-count denominator.
Pr(K=0)Pr(K=1)=24−r4r=2
Model
Model
The simplified probability ratio yields a linear equation whose admissible solution is eight insured items, leaving 19 uninsured items.
4r=48−2r,r=8
Compute
Compute
Count samples with exactly two insured items by choosing two from the eight insured items and two from the 19 uninsured items, then divide by all four-item samples.
Pr(K=2)=(427)(28)(219)=975266=0.27282
Answer
Answer
The required sampling probability rounds to 0.27.
0.27(C)
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Original variant: premium labels in an inspection sample
A warehouse has 30 sealed packages, 10 bearing a premium label and 20 bearing a standard label. Five packages are selected uniformly without replacement. Find the probability that exactly two selected packages bear premium labels.
A 0.1600
B 0.2400
C 0.3000
D 0.3600
E 0.4800
Variant answer in brief
There are C(10,2)C(20,3) favorable subsets among C(30,5) possible samples, giving 0.359985, choice D.
Setup
Setup
Every five-package subset is equally likely under sampling without replacement, so the denominator is the number of five-package subsets.
Nall=(530)
Model
Model
A favorable sample chooses two of the ten premium packages and three of the twenty standard packages.
Nfav=(210)(320)
Compute
Compute
Dividing the favorable subset count by the total subset count gives 0.359985, which rounds to 0.3600.
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