Independent solution

How to solve this Multinomial Distribution question

Setup

Setup

Convert the nested classification rates into three mutually exclusive outcome probabilities: outside the broad category, inside it but outside the target subtype, and inside the target subtype.

p0=0.80p_0=0.80
p1=0.20(1−0.08)=0.184p_1=0.20(1-0.08)=0.184
p2=0.20(0.08)=0.016p_2=0.20(0.08)=0.016

Model

Model

If the stopping time is n and the broad-category total is c, the terminal observation must be target subtype. Among the preceding n-1 observations, the three category counts are determined.

(n−c) of type 0,(c−5) of type 1,4 of type 2(n-c)\text{ of type }0,\qquad(c-5)\text{ of type }1,\qquad4\text{ of type }2
N=n⟹observation n is type 2N=n\Longrightarrow\text{observation }n\text{ is type }2

Compute

Compute

Count the admissible orders of the first n-1 outcomes, multiply by their category probabilities, and append the terminal target-subtype probability.

pN,C(n,c)=(n−1)!4!(c−5)!(n−c)!(2125)4(23125)c−5(45)n−c(2125)p_{N,C}(n,c)=\frac{(n-1)!}{4!(c-5)!(n-c)!}\left(\frac{2}{125}\right)^4\left(\frac{23}{125}\right)^{c-5}\left(\frac45\right)^{n-c}\left(\frac{2}{125}\right)
pN,C(n,c)=(n−1)!4!(c−5)!(n−c)!(2125)5(23125)c−5(45)n−cp_{N,C}(n,c)=\frac{(n-1)!}{4!(c-5)!(n-c)!}\left(\frac{2}{125}\right)^5\left(\frac{23}{125}\right)^{c-5}\left(\frac45\right)^{n-c}

Answer

Answer

The terminal constraint and multinomial coefficient select the first listed probability function.

choice A\boxed{\text{choice A}}