This Exam P sample reference tests Hypergeometric Distribution. At most two defective selections fails only when all three defectives are among the four selected items. That excluded event has probability 1/30, leaving 29/30 = 0.966667 and choice E.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis repeats the population defective fraction 3/10 = 0.30 instead of calculating a probability for the four-item selection.
BThe excluded event has probability 1/30 = 0.0333, not 0.33; this choice reflects both taking the complement event and losing a decimal place.
CUsing the ratio 2/3 because two of three defectives are allowed ignores the random selection mechanism and is not a hypergeometric probability.
DTreating the four draws as independent Bernoulli trials with defect probability 0.30 gives P(X ≤ 2) = 0.9163, about 0.92; sampling is actually without replacement.
Original practice · fully worked
Original variant: firmware audit sample
A warehouse tray contains 12 control modules, four of which require a firmware update. A technician selects five modules uniformly without replacement for an audit. Calculate the probability that at most three of the selected modules require the update.
A 0.333
B 0.010
C 0.667
D 0.955
E 0.990
Variant answer in brief
The only way to exceed three update-required modules is to select all four of them. That event has probability 1/99, so the requested probability is 98/99 = 0.989899 and choice E.
Setup
Setup
Let Y be the number of update-required modules among five selected from four update-required and eight current modules.
Y∼Hypergeometric(N=12,K=4,n=5)
Model
Model
The complement of at most three is selecting all four update-required modules, together with one current module.
Pr(Y≤3)=1−Pr(Y=4)
Compute
Compute
Evaluate the single complement count and subtract it from one.
Pr(Y=4)=(512)(44)(18)=7928=991
Pr(Y≤3)=9998=0.989898…
Answer
Answer
The audit contains at most three modules needing an update with probability approximately 0.990.
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.
