Need an example of a Binomial Distribution based on a questi

Need an example of a Binomial Distribution based on a question like this

A Box has 10 red and 15 blue balls. Draw balls one by one with replacement until you get 5 blue balls. Find the probability that you need at least 8 such draws?

Students Comment

I so far have (PX>=8) = 1 - P(X<8)

I have 1-.1003-.0668-.0420 as an answer. I think my solution is too high.

Solution

Given that a box has 10 red balls and 15 blue balls.

We have to draw a balls with replacement until get 5 blue balls.

Here random variable X is number of failuare before 8th trial.

r = number of successes = 5

The random variable X follows negative binomial distribution with parameter r = 5 and p = 2/3.

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

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

We have to find the probability that at least 8 such draws that means we have 5 successes in 8 draw and 3 failuares.

So we have to find the P(X<3) = P(X=0)+P(X=1)+P(X=2)+P(X=3)

P(X=0) = (0+5-1 C 5-1) * (2/3)^5 * (1/3)^0 = 0.1317

P(X=1) = (1+5-1 C 5-1) * (2/3)^5 * (1/3)^1 = 0.2195

P(X=2) = (2+5-1 C 5-1) * (2/3)^5 * (1/3)^2 = 0.2195

P(X=3) = (3+5-1 C 5-1) * (2/3)^5 * (1/3)^3 = 0.1707

P(X<3) = 0.7414

This is the probability for 8 draws.