Transition Matrix for a betting game

[ha9981]

Suppose that a casino introduces a game in which a player bets $1 and can
either win $2 or lose it, both with equal chances. The game ends when the player runs out
of money, or when he wins $4.
(a) Build a transition matrix for the game, and show that it is not a regular transition
matrix.
(b) Find the long term expected payoff to the player, and explain why the game is
pro profitable (or not) for the Casino.

My Attempt:

a) P =
0.5 0.5
0.5 0.5

I don't feel this is right because there is a lot of extra information in the question that seems wasted. My attempt to incorporate game ending at $4.

P =
0.5 0.5 0
0.5 0.5 1


I can't even get to part B as I am struggling at the transition matrix, if someone could guide me to a similar example because my textbook lacks here.

 

[fresh_42]

This should be a Markov process, but we could as well list all possible paths. It is always a good idea to draw a graph.

 

[statdad]

Suppose that a casino introduces a game in which a player bets $1 and can
either win $2 or lose it, both with equal chances. The game ends when the player runs out
of money, or when he wins $4.
(a) Build a transition matrix for the game, and show that it is not a regular transition
matrix.
(b) Find the long term expected payoff to the player, and explain why the game is
proprofitable (or not) for the Casino.

My Attempt:

a) P =
0.5 0.5
0.5 0.5

I don't feel this is right because there is a lot of extra information in the question that seems wasted. My attempt to incorporate game ending at $4.

P =
0.5 0.5 0
0.5 0.5 1I can't even get to part B as I am struggling at the transition matrix, if someone could guide me to a similar example because my textbook lacks here.

How much does the player start with? Nothing? Two dollars? Once you know that list out as the states the amounts of money the person could have after each bet, whether win or lose, then assign probabilities.