Question

Assume that a fair six-sided die is rolled 9 times, and the roll is called a success if the result is in {1,2}{1,2}.

What is the probability that there are exactly 4 successes or exactly 4 failures in the 9 rolls?

Answer 1

Here' the answer to the question. Let me know in case you've doubts.

We will use the binomial distribution pdf function to solve the problem

First lets define probability of success first.

On the 6 numbers of dice, event of success is getting 1 or 2

P(X) = P(getting 1 or 2 out of the 6 numbers on dice) = 2/6 = 1/3

So, p = 1/3

Therefore, probability of failure is 1- P(success) = 1-p = 1-1/3 = 2/3

Hence, out of the 9 rolls P(getting 4 exactly success or exactly 4 failures)

= P(X=4 success) + P(X=4 failures)

= 9C4 * (1/3)^4 *(2/3)^5 + 9C4 * (2/3)^4 *(1/3)^5

= 0.3069

Answer: P(exactly 4 success or exactly 4 failures on 9 die rolls) = 0.3069