This Exam P sample reference tests Continuous Uniform Distribution. A wait above three minutes occurs only for an arrival in the first two minutes of the five-minute closure, so each vehicle has probability 2/15. Independence gives 1-(13/15)²=56/225=0.2489, choice A.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BAdding the two marginal probabilities gives 4/15=0.2667, nearest 0.27. This double-counts the case in which both vehicles arrive in the critical interval.
CUsing a three-minute critical interval gives individual probability 3/15=0.20 and union 1-0.80 squared=0.36. An arrival with exactly three minutes remaining does not exceed three minutes, and the interval length is two.
DThe value 0.40 adds two erroneous 0.20 marginal probabilities and also omits their overlap.
EUsing the full five-minute closure gives individual probability 5/15 and union 1-(2/3) squared=5/9=0.5556, nearest 0.56. Late arrivals during the closure wait three minutes or less.
Original practice · fully worked
Original variant: exactly two reports in a secure window
Four inspection reports are submitted independently at times chosen uniformly over an eight-hour shift. Direct upload is available during one fixed 1.5-hour secure window. Calculate the probability that exactly two reports arrive during that window.
A 0.0352
B 0.1393
C 0.1875
D 0.4358
E 0.5642
Variant answer in brief
Each report lands in the secure window with probability 1.5/8=3/16. The number arriving there is binomial with size 4, so the probability of exactly two is C(4,2)(3/16)²(13/16)²=0.1393, choice B.
Setup
Setup
Convert the secure window's length into one report's arrival probability.
p=81.5=163
Model
Model
Independence makes the number of reports in the window binomial.
K∼Binomial(4,163)
Compute
Compute
Choose the two in-window reports and require the other two to fall outside.
Pr(K=2)=(24)(163)2(1613)2
Pr(K=2)=0.1392517
Answer
Answer
Exactly two reports arrive in the secure window with probability about 0.1393.
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.
