rolling two homogenous dice..

Question

Roll two homogenous dice until a total of 7 appears for the first time. Find the probability that the number of rolls needed is odd? (using one of the special distribution)

As I know we should use geometric distribution if I'm not wrong since they ask for the total of 7 appears at the first time.

My question is:

1. why they mentioned that the dices are homogenous?
2. how can we use the geometric distribution and we don't have the probability?

Thanks,

Answer 1

The probability of a total 7 is $\frac{1}{6}$ thus the special pmf (geometric) is the following

$$\mathbb{P}[X=x]=\frac{1}{6}\cdot\left( \frac{5}{6} \right)^{x-1}$$

$x=1,2,3,4,\dots$

thus

$$\mathbb{P}[X\text{ is Odd}]=\frac{6}{11}$$