Worked solution · Exam P
This Exam P sample reference tests Sampling Without Replacement. A match must occur by draw five, and reaching that maximum requires the first four draws to have four different colors. The sequential probability is (6/7)(4/6)(2/5)=8/35=0.2286, choice E.
Independent solution
Setup
Use the four available colors to identify the latest possible first match.
Model
After each new color, count the remaining socks whose colors have not yet appeared.
Compute
Multiply the three without-replacement continuation probabilities.
Answer
The maximum draw count is required with probability about 0.2286.
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Original practice · fully worked
A sealed rack contains five green tags and seven amber tags. A technician exposes tags one at a time in a uniformly random order and stops upon seeing the first green tag. If T includes the stopping exposure, determine E[T].
The five green positions form a uniform five-subset of the twelve positions. The expected minimum selected position is (12+1)/(5+1)=13/6=2.1667, choice C.
Setup
View the random exposure order as five marked green positions among twelve positions.
Model
For K marked positions chosen uniformly from N positions, the expected first marked position is (N+1)/(K+1).
Compute
Insert the twelve total positions and five green positions.
Answer
The expected stopping exposure is approximately 2.167.
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