Worked solution · Exam P
This Exam P sample reference tests Discrete Random Variables. This problem reconstructs a single probability cell from overlapping count ranges. First p_3=0.260-0.250-0.002=0.008, then p_2=0.036-0.008=0.028, so choice D.
Independent solution
Setup
Let p_k be the probability of exactly k events, and let p_{4+} collect all counts of four or more.
Model
Subtract the first range and the upper tail from the at-least-one probability to isolate the three-count cell.
Compute
Remove the recovered three-count probability from the second overlapping range.
Answer
The probability of exactly two events is 0.028.
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Original practice · fully worked
A quality record can carry flag A, flag B, both flags, or neither. The probabilities of flag A and flag B are 0.42 and 0.35, respectively. The probability of carrying exactly one flag is 0.53. Calculate the probability that a record carries neither flag.
If x is the both-flags probability, exactly-one probability is 0.42+0.35-2x=0.53, giving x=0.12. The union is then 0.65, so the neither probability is 0.35 and choice C.
Setup
Let x be the probability that both flags are present.
Model
Adding the two marginal probabilities counts both-flags records twice, so subtract two copies of x to obtain exactly one flag.
Compute
Solve for the intersection, then use inclusion-exclusion and a complement.
Answer
The probability of neither flag is 0.35.
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