So lets say I am rolling a dice untill I get three ones What

So lets say I am rolling a dice untill I get three one\'s. What distribution would I use to find the probability of rolling the dice at least six times?

Solution

Given that rolling a dice untill I get three one\'s.

Let X be a random variable number of failuare preceeding third throw of die appear one.

P(success) = 1/6 = 0.1667

P(failuare) = 1 - 0.1667 = 0.8333

X follows Negative binomial distribution with parameter r and P.

where r is number of successes =3

P=probability of success appearing 1

P(X>=6) = 1 - P(X<6)

The p.m.f. for Negative binomial distribution is ,

P(X =x) = (x+r-1 C x) * p^x * q^r

we have given x+r = 6

And we have to find P(X>=6)