Problems regarding probability A fair die is alternately thrown by two persons . The first one wins if one dot appears and the second one wins if 2-dots or 3-dots appear. The first one starts throwing the die. What is the probability that the first one ultimately wins ?

**HINT**

Denote by $W$ (or $L$) the event that the first one wins (or loses). $W$ consists of some possibilities:

* in one rolls is $1/6$
* in 3 rolls is $5/6$ for rolling not 1, $4/6$ for the second rolling not 2 nor 3, and $1/6$ for rolling 1 on the 3rd roll
* all subsequent ones just add extra factors of $5/6$ and $4/6$ for another iteration.



Now clear pattern emerges: $$ \mathbb{P}[W] = \frac16 + \frac 56\frac46\frac16 + \left(\frac 56\frac46\right)^2\frac16 + \ldots $$ Can you sum the geometric series?

You should also compute $\mathbb{P}[L]$ that way and check $\mathbb{P}[W]+\mathbb{P}[L]=1$...