This Exam P sample reference tests Event Independence. Each event has probability one half and each pair has intersection probability one quarter, but the triple intersection is empty rather than one eighth. Thus the events are pairwise but not mutually independent, choice A.
Check pairwise independence and mutual independence separately. The three marginal probabilities are all one half.
Pr(A)=Pr(B)=Pr(C)=21
Model
Model
Each pairwise intersection has probability one quarter, equal to the product of the corresponding marginal probabilities. Thus every pair is independent.
Pr(A∩B)=Pr(A∩C)=Pr(B∩C)=41
Compute
Compute
Mutual independence would require a triple-intersection probability of one eighth, but the actual triple intersection is empty. Pairwise independence therefore holds without mutual independence.
Pr(A∩B∩C)=0=81
Answer
Answer
Every pair is independent, while the three events together are not.
pairwise independent, not mutually independent(A)
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BMutual independence fails because the triple intersection has probability zero rather than the required one eighth.
CThis choice is incompatible with the three pairwise checks, each of which gives an intersection probability equal to one quarter.
DThis choice is incompatible with the fact that all three pairs pass the independence test.
EThis choice is incompatible with the fact that every pair has intersection probability equal to the product of its marginal probabilities.
Original practice · fully worked
Original variant: two coins and a matching event
Two fair coins are tossed. Let A be the event that the first coin is heads, B the event that the second coin is heads, and C the event that the two coins match. Determine the correct independence relationship among A, B, and C.
A Pairwise independent but not mutually independent
B Mutually independent
C Only A and B are independent
D Exactly two pairs are independent
E No pair is independent
Variant answer in brief
All three marginal probabilities are one half and every pairwise intersection has probability one quarter. The triple intersection has probability one quarter, not one eighth, so choice A holds.
Setup
Setup
The three events each have probability one half under the four equally likely coin outcomes.
Pr(A)=Pr(B)=Pr(C)=21
Model
Model
Each pairwise intersection has probability one quarter, exactly the product of the two marginal probabilities, so all three pairs are independent.
Pr(A∩B)=Pr(A∩C)=Pr(B∩C)=41
Compute
Compute
All three events occur together only on the two-heads outcome, with probability one quarter. This differs from the product of the three marginal probabilities, one eighth.
Pr(A∩B∩C)=41=81
Answer
Answer
The events are pairwise independent but fail mutual independence.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
