I have the following problem:
*A gambler goes to a gambling hall. The gambler decides to gamble on two machines in the following way. Every odd turn, the gambler uses machine A, while every even turn, the gambler uses machine B. The machines have the following characteristics:
• Every turn costs 1 euro.
• Machine A yields 2 euro with probability p or 0 euro with probability 1 − p, 0 < p < 1;
• Machine B yields 2 euro with probability r or 0 euro with probability 1 − r, 0 < r < 1.
What is the one-state transition probabilities matrix?*
I am struggling to create a matrix in which I can distinguish between probabilities p and r as if depending on the state the probability will be one or the other.
Say the gambler's initial bankroll is $n$. At any future time, his bankroll is $n+k$ for some integer $k$. If $k$ is even, then next play will be on machine A. If $k$ is odd, it will be on machine B.