Question

Please make sure to display your thought process? It is imperative to be able to follow how the answer was deduced. Please be as thorough as possible. Please address all parts of this question.

You roll a fair die (dice) 5 times. Calculate the following probabilities and indicate which probability distribution model is appropriate in each case.
(a) What is the probability of rolling the first 6 on the fifth roll?
(b) What is the probability of rolling 6 exactly three times?

Answer 1

(a)

Probability of rolling a 6 on roll of a fair dice is p = 1/6

Let X be the number of rolls required to get a 6. Then X follows Geometric distribution with parameter p = 1/6

From PMF of Geometric distribution,

P(X = 5) = (1 - 1/6)^5-1 * (1/6) = (5/6)⁴ * (1/6) = 0.0804

Probability of rolling the first 6 on the fifth roll = 0.0804

(b)

Let X be the number of 6 you get on 5 rolls of dice. Then X follows Binomial distribution with parameter n = 5 and p = 1/6

P(X = 3) = ⁵C₃  * (1/6)³ * (1 - 1/6)²

= 10 * (1/6)³ * (5/6)²

= 0.03215