Probability of winning (dice) Given a simple dice $\{1,2,3,4,5,6\}$ : A win is defined as a sequence of $1 \to 2 \to 3$.
Every result which "breaks" the sequence (e.g. $1\to 2 \to 1$), is forcing us to start all over again.
What is the minimum number of rounds needed to have the chance for winning (at least once) equals $0.25$?
 A: Hints (preassuming that $1\rightarrow2\rightarrow1\rightarrow2\rightarrow3$ represent $3$ loosing rounds) :


*

*The probability that a round ends up in winning is $p:=\frac16\frac16\frac16$.

*If $n$ rounds are played then the probability of no winning at all is $(1-p)^n$.

A: Off the top of the head, below are two approaches I would consider.
Direct way. Draw an "execution" tree to see what's going on. You start at "root" node that has six edges leading out so that $i$th edge corresponds to throwing number $i$; the children are defined in the same way. So you get an infinite tree. Now notice that the path that goes from the root node along the edges labeled $1$, $2$, and $3$ is a "good" path; similarly, all the paths $i$,$1$,$2$,$3$ for $i>1$ are "good" paths; and so on. Of course each path corresponds to a sequence of dice throws and the probability of finishing the game in $n$ throws is precisely the sum of probabilities corresponding to "good" paths of length at most $n$. 
Think about how to compute these probabilities.
Random walk. It seems you could view this as a random walk on a graph that has $\{1,2,3,4\}$ as a vertex set. The transition matrix should reflect rules of the game:


*

*if you are in state $1$ and you throw a $1$ then move to state $2$, otherwise stay at state $1$;

*if you are in state $2$ and you throw a $2$ then move to state $3$, otherwise go to state $1$;

*if you are in state $3$ and you throw a $3$ then move to a "final" state, otherwise go to state $1$.


You need to think about what the final state will be (and if you even need a special final state). My guess is that you could model this as a discrete Markov chain that will turn out to have nice properties and you will be able to use one of the standard theorems that will help you answer the question. 
