Worked solution · Exam P
This Exam P sample reference tests Geometric Distribution. A positive admission count is geometric with parameter 2/3. Enumerating the four ways that five employees can generate at most two admissions gives probability 0.78279, choice E.
Independent solution
Setup
Translate the cost threshold into a count threshold and recover the geometric parameter.
Model
For one employee, record the probabilities needed for aggregate counts zero, one, and two.
Compute
Add the disjoint aggregate configurations with total admission count at most two.
Answer
The required probability rounds to 0.78.
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Original practice · fully worked
The number N of incidents reaching a network hub in one hour is Poisson with mean 0.8. Each incident independently produces a positive geometric number of alerts with mean 2, and alert counts from different incidents are independent. Find the probability that the hub receives at most one alert during the hour.
At most one alert means either no incidents, or one incident producing exactly one alert. These probabilities sum to exp(-0.8)(1+0.8/2)=0.62906, choice D.
Setup
Recover the geometric success parameter from its mean.
Model
Because every incident creates at least one alert, list the only cases yielding total zero or one.
Compute
Evaluate the two disjoint compound-count cases.
Answer
The low-tail probability is approximately 0.62906.
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