Original practice · fully worked
Original variant: downlink timing within a pass
Telemetry downlinks arrive according to a homogeneous Poisson process. During a four-hour satellite pass, exactly six downlinks are recorded. Conditional on that total, calculate the probability that exactly two of the downlinks occurred during the first hour.
- A
0.234 - B
0.251 - C
0.297 - D
0.356 - E
0.466
Answer and calculation
Given six arrivals in four hours, each arrival independently falls in the first hour with probability 1/4. The resulting binomial probability is 0.29663, so choice C is correct.
Setup
Setup
Condition on the observed total of six arrivals over the complete four-hour pass.
Model
Model
Conditional on the total, homogeneous Poisson arrival times are uniformly distributed across the pass. Each arrival therefore lands in the first one-quarter of the interval with probability one-quarter.
K∣N(4)=6∼Binomial(6,41) Compute
Compute
Evaluate the conditional binomial mass at two.
Pr(K=2∣N(4)=6)=(26)(41)2(43)4 Pr(K=2∣N(4)=6)=0.2966308594… Answer
Answer
The conditional probability rounds to 0.297.
0.297(C) 