Is this a negative binomial?

Amer

New Member
#1
Is this a negative binomial? I understand the solutions. But why am I not getting the right answer using the negative binomial formula?

Problem:
What is the probability that it takes exactly 10 rolls to observe 8 sevens on 2 fair six sided dice.

My work:
using the negative binomial formula (x+r-1)C(x) (P)^x (1-P)^r
(10+8-1)C(10) (1/6)^10 (5/6)^8 = .000074802

How the solutions solve it:
(9)C(7) * (1/6)^7 * (5/6)^2 * (1/6) = .000014884

Thanks
 

BGM

TS Contributor
#2
The suggested solution seems implying exactly 9 rolls but not 10...

Anyway, the bottom line is:

1. For the negative binomial, you are required that the last roll is success,
while the order of success in the previous rolls are not important, and hence
you get the binomial coefficient. So in your formula, you are getting the
7th success exactly on the 10th roll. And thus you miss some cases where
you get the 7th success before the 10th roll, which shall be included by the
question.

2. For the binomial distribution, the order of success in the 10 rolls are not
important. They can be appear in any order, but you just required that there
are 7 successes in total.