This Exam P sample reference tests Sampling Without Replacement. The expected number of white chips transferred is 2(4/9)=8/9. The destination therefore has expected white count 5+8/9 among 11 chips, giving 53/99=0.5354 and choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis assumes no white chip is transferred and uses 5/11=0.4545 for the destination draw.
CThis assumes exactly one white chip is transferred and reports 6/11=0.5455 instead of averaging over all transfer outcomes.
DThis reports the destination's original white proportion 5/9=0.5556, ignoring both the transferred composition and the increase to 11 chips.
EThis assumes both transferred chips are red, then looks at the wrong container and reports the source's remaining white proportion 4/7=0.5714.
Original practice · fully worked
Original variant: two rare books after relocation
A storage cart holds 10 books, 4 rare and 6 ordinary. Three books are selected uniformly without replacement and moved to a display shelf that already holds 2 rare and 6 ordinary books. Two books are then selected uniformly from the 11 books on the display shelf. Calculate the probability that both selected books are rare.
A 0.01818
B 0.05455
C 0.06909
D 0.10909
E 0.18182
Variant answer in brief
Averaging the number of rare pairs on the display shelf over the hypergeometric relocation gives 19/5 expected rare pairs. Dividing by C(11,2)=55 gives 19/275=0.06909 and choice C.
Setup
Setup
Let K be the number of rare books among the three relocated books.
Pr(K=k)=(310)(k4)(3−k6),k=0,1,2,3
Model
Model
Given K, the display holds 2+K rare books among 11, so the two-book rare probability is hypergeometric.
Pr(RR∣K=k)=(211)(22+k)
Compute
Compute
Average the conditional pair probabilities across the four relocation outcomes.
Pr(RR)=61551+21553+103556+3015510
Pr(RR)=27519=0.0690909091
Answer
Answer
The probability of selecting two rare books is approximately 0.06909.
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.
