Creating n Equations in n Variables for Probability Problems

A fair coin is flipped repeatedly until either $HH$ or $TT$ occurs.

What is the probability that $HH$ occurs before $TT$?

A token starts at 2 on the number line

At every step it moves one unit right with probability $2/3$ and one unit left with probability $1/3$. The process continues until it reaches 0 or 3.

What is the probability that it reaches 3 first?

A game can be in state $A$ or state $B$.

From $A$:

• with probability $1/3$, you win;

• with probability $2/3$, you move to $B$.

From $B$:

• with probability $1/4$, you lose;

• with probability $3/4$, you return to $A$.

Starting from $A$, find the probability of eventually winning.