Worked solution · Exam P
This Exam P sample reference tests Binomial Distribution. The count is binomial with n=18 and p=0.15. Since (n+1)p=2.85 is not an integer, its unique mode is floor(2.85)=2, so choice C is correct.
Independent solution
Setup
Model the number of positive results as a binomial count.
Model
A binomial distribution has unique mode floor((n+1)p) when (n+1)p is not an integer.
Compute
Take the integer below 2.85.
Answer
The most likely count is 2.
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30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
Error diagnosis
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Original practice · fully worked
The number X of flagged items in ten independent inspections is binomial with an unknown flag probability. The probability of exactly four flags is 1.5 times the probability of exactly three flags. Calculate the mode of X.
The adjacent-mass ratio gives 7p/[4(1-p)]=1.5, so p=6/13. Then (10+1)p=66/13 lies between 5 and 6, making 5 the unique mode and choice C correct.
Setup
Use the ratio of adjacent binomial masses to recover the unknown probability.
Model
Solve the ratio equation for p.
Compute
Apply the binomial mode rule after inferring the parameter.
Answer
The most likely number of flagged items is 5.
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