Independent solution

How to solve this Negative Binomial Distribution question

Setup

Setup

The trial count for the second damage event has the displayed negative-binomial mass on integers starting at 2.

P(N=n)=(n−1)(0.4)2(0.6)n−2,n≥2P(N=n)=(n-1)(0.4)^2(0.6)^{n-2},\qquad n\ge2

Model

Model

Compare consecutive masses through their ratio to locate where the sequence changes from increasing to decreasing.

P(N=n+1)P(N=n)=nn−1(0.6)\frac{P(N=n+1)}{P(N=n)}=\frac{n}{n-1}(0.6)

Compute

Compute

The ratio from count 2 to 3 is 1.2, while the ratio from 3 to 4 is 0.9. Thus the peak occurs at 3.

ratio at n=2=1.2>1\text{ratio at }n=2=1.2>1
ratio at n=3=0.9<1\text{ratio at }n=3=0.9<1

Answer

Answer

The modal trial count is 3, corresponding to choice B.

3(B)\boxed{3\quad\text{(B)}}