Worked solution · Exam P
This Exam P sample reference tests Counting Principles. Choose the repeated face, the different face, and the position of the different die. This gives 6(5)(3)=90 favorable ordered outcomes out of 216, so the probability is 5/12, approximately 0.417 and choice B.
Independent solution
Setup
The three distinguishable dice generate 6 cubed equally likely ordered outcomes.
Model
An outcome with exactly one pair is determined by its repeated value, its different value, and the location of the different value.
Compute
Divide the favorable ordered outcomes by the complete sample space.
Answer
The requested probability rounds to 0.417.
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Original practice · fully worked
A service desk receives seven distinct repair tickets labeled A through G and schedules them in a uniformly random order. Calculate the probability that ticket A is scheduled earlier than both B and C, while ticket D is scheduled later than E.
Among the relative orders of A, B, and C, ticket A is earliest in two of the six orders, giving probability 1/3. Independently, D is later than E with probability 1/2. Their product is 1/6, approximately 0.1667 and choice B.
Setup
Focus on relative order within the two disjoint ticket groups; the positions of the remaining tickets do not affect either requirement.
Model
Each member of the three-ticket group is equally likely to be earliest, and either order of D and E is equally likely.
Compute
Relative orders of disjoint labeled groups are independent, so multiply the two symmetry probabilities.
Answer
Both relative-order conditions hold with probability about 0.1667.
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