Gambler's Ruin

First-step analysis for the probability and expected time to hit one of two absorbing boundaries.

1. The idea

A gambler starts with $i$ dollars and bets $1$ dollar repeatedly, winning with probability $p$ and losing with probability $q = 1 - p$ , stopping at $0$ (ruin) or $N$ (target). First-step analysis conditions on the outcome of the very next bet to build a recursion.

For a fair game ( $p = q = 1/2$ ), the probability of reaching $N$ before $0$ starting from $i$ is simply $i / N$ — proportional to the starting fraction of the target — and the expected time to absorption is $i (N - i)$ .

For a biased game ( $p \neq = 1/2$ ), the ruin probabilities involve the likelihood ratio $r = q / p$ : $P (hit N first ∣ i) = \frac{1 - r ^{i}}{1 - r ^{N}}$ .

2. Why it matters

First-step analysis (condition on the first move, write a recursion, solve using boundary conditions) is a general-purpose technique for any absorbing Markov chain, not just casino games — it appears in inventory, ruin, and pricing recursions.

3. Core mathematics

Setup

0 \leq i \leq N, p = P (win each bet), q = 1 - p

Fair game hitting probability

P (hit N before 0 ∣ i) = \frac{i}{N}

Fair game expected time

E [time to absorption ∣ i] = i (N - i)

Biased game hitting probability

P (hit N before 0 ∣ i) = \frac{1 - r ^{i}}{1 - r ^{N}}, r = \frac{q}{p} \neq = 1

Recursion (first-step)

h (i) = p h (i + 1) + q h (i - 1), h (0) = 0, h (N) = 1

4. How to recognize it

Think of this when…

A process with two absorbing boundaries (bankruptcy vs. target wealth, or two competing states)
"Starting with $i$ , what is the probability of reaching $N$ before $0$ ?"
Repeated $\pm 1$ bets or steps with fixed win probability $p$
Requests for expected time until absorption

5. Worked example

Worked example

A gambler starts with $$4$ and plays a fair game ( $p = 1/2$ ) against a target of $$10$ , betting $$1$ each round. Find the probability of reaching $$10$ before going broke, and the expected number of rounds played.

1
This is fair gambler's ruin with $i = 4$ , $N = 10$ .
2
Use the fair-game hitting probability formula.
$P (reach 10 first) = \frac{i}{N} = \frac{4}{10} = 0.4$
3
Use the fair-game expected duration formula.
$E [rounds] = i (N - i) = 4 \times 6 = 24$
Answer: Probability of reaching $10 first is $0.4$ ; expected number of rounds is $24$ .

6. Common mistakes

Using the fair-game formula $i / N$ when the game is biased ( $p \neq = 1/2$ )
Mixing up which boundary is $0$ vs. $N$ , flipping the probability
Forgetting boundary conditions $h (0) = 0, h (N) = 1$ when setting up the recursion from scratch
Applying $i (N - i)$ for expected time in a biased game, where it does not hold

7. Quick check

In a fair gambler's ruin with $i = 3$ , $N = 8$ , what is the probability of reaching $N$ before $0$ ?

What recursion underlies the gambler's ruin hitting probability $h (i)$ ?

8. Practice

2 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Recursive Expectation / First-Step Analysis Conditioning on the First Event

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