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
Answer and calculation
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.
0.06909(C) 