expected value and variance

a games is played on a board marked off into a line of four squares. a player starts on the square numbered 2. on each turn, a six-sided die is thrown, and the players moves one square to the left if the die comes up 5 or 6, and one square to the right otherwise. the game ands when the player reaches either and of the board (square 1 or square 4).

[1][2][3][4]

a/ let x is the numbers of moves until the game ends. find the frequency function of x
b/ find E(x)
c/ find Var(x).

david said:

a games is played on a board marked off into a line of four squares. a player starts on the square numbered 2. on each turn, a six-sided die is thrown, and the players moves one square to the left if the die comes up 5 or 6, and one square to the right otherwise. the game ands when the player reaches either and of the board (square 1 or square 4).
[1][2][3][4]
a/ let x is the numbers of moves until the game ends. find the frequency function of x
b/ find E(x)
c/ find Var(x).

Click to expand...

I assume that moving from \(\displaystyle \boxed{2}\text{ to }\boxed{1}\) is moving one place to the left.
Now the probability function depends upon if n, the number of moves to win, is even or odd.

If n is odd then \(\displaystyle P(X=n)=\left( {\dfrac{2}{3}} \right)^{\frac{{n - 1}}{2}} \left( {\dfrac{1}{3}} \right)^{\frac{{n + 1}}{2}} \)

If n is even then \(\displaystyle P(X=n)=\left( {\dfrac{2}{3}} \right)^{\frac{{n+2 }}{2}} \left( {\dfrac{1}{3}} \right)^{\frac{{n -2}}{2}} \).

The player can win in one move to the left.
The player can win in two moves to the right.
The player can win in three moves like this \(\displaystyle RLL,~\boxed{3}\boxed{2}\boxed{1}\).
i.e. moves to three, then two, and then one.

Now it is up to you to finish.