Question

A fair six-sided die is rolled repeatedly until the third time a 6 is rolled. Let X denote the number of rolls required until the third 6 is rolled. Find the probability that fewer than 5 rolls will be required to roll a 6 three times.

Answer 1

Negative Binomial Distribution:

The random variable X denote the trial at which the r^th success occurs, where r is a fixed integer.

X has a negative Binomial(r,p) distribution

For the Given problem

X : Number of rolls required until the third 6 is rolled

Success is rolling a six; third six is rolled i.e third success i.e r =3

p : Probability of success i.e Probability of rolling a 6 = 1/6

Therefore, substituting r=3 and p=1/6 in the above shown probability distribution formula

i.e

Probability that fewer than 5 rolls will be required to roll a 6 three times = P(X<5)

P(X<5) = P(X=3)+P(X=4)

P(X<5) = P(X=3)+P(X=4) = 0.004630+0.011574 = 0.016204

Probability that fewer than 5 rolls will be required to roll a 6 three times = 0.016204