When studying the Gambler’s Ruin problem, beginners often confuse the state numbers with the probabilities attached to those states.
States Represent the Current Situation
Suppose Player A currently owns some amount of money.
- State 0 means Player A has no money.
- State 1 means Player A has $1.
- State 2 means Player A has $2.
- …
- State N means Player A has all the money.
The numbers 0, 1, 2, …, N are simply labels for the possible states of the game.
Probabilities Are Attached to States
For each state i, we define:
This means:
“Probability that Player A eventually wins when starting from state i.”
Examples:
= Probability of winning when starting with $0.
= Probability of winning when starting with $1.
= Probability of winning when starting with $2.
Why is P₀ = 0?
If Player A starts with no money, the game is already lost.
Therefore:
Why is Pₙ = 1?
If Player A starts with all N dollars, the game is already won.
Therefore:
Example: N = 4
Possible states:
- State 0 → $0
- State 1 → $1
- State 2 → $2
- State 3 → $3
- State 4 → $4
Associated probabilities:
Known values:
Unknown values:
These are the probabilities we solve for.
Why Use the Same Subscript?
The subscript tells us the starting state.
For example:
means the probability of eventual victory when starting from state 3.
Key Takeaway
The numbers:
represent states.
The quantities:
represent probabilities attached to those states.
In short:
- State i tells us where we are.