Gambler’s Ruin

Start with i dollars. Each play wins $1 with probability p and loses $1 with probability q = 1 − p. What is the probability of reaching N before hitting 0?

Goal N Starting dollars i Win probability p0.50

Probability of reaching N first

40.00%

q = 0.50

$0RUIN

$4START

$10GOAL

Run a simulation and compare it with the exact answer.

How to derive the formula

① Rearrange the recurrence pPᵢ₊₁ − Pᵢ + qPᵢ₋₁ = 0

② Solve the difference equation The characteristic roots are 1 and q/p.

③ Use the two boundaries Enforce P₀=0 and Pₙ=1 to determine the constants.

Final answer

If p = q = 1/2:

Pᵢ = i / N

Otherwise:

Pᵢ = [1 − (q/p)ⁱ] / [1 − (q/p)ᴺ]