Question

Imagine rolling a fair 6-sided die until we get a six. We know that the probability that this occurs on the nth roll is (5/6)n−1·(1/6). Now describe:

- the infinite sample space of the experiment

- the probability function for this experiment

- Show that your probability function satisfies Pr(Ω) =1

Describe how you obtained your answers.

Answer 1

Let, X = total number of rolling a fair 6-sided die until we get a six.

Therefore the sample space = = { 1, 2, 3, ..... }

The probability of getting 6 when we roll a fair die is (1/6)

Therefore, the probability distribution of X is given by :

n = 1, 2, 3....

and P( X = n ) = 0 , otherwise.

Here we want to find P( )

Here we use geometric series as  

for | x | < 1.