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Mathematical Calculation of Risk of Ruin For a Fixed EV

Applies to a fixed expectation with a fixed bet of 1 unit

let win = probability of win
let loss = probability of loss
let tie = probability of tie (win + loss + tie = 1)
let bank = number of units gambler is willing to risk
let profit = number of units gambler hopes to win
let goal = bank + profit = number of units gambler wishes to achieve
let S_bank = probability of succeeding with bank units

S_goal = 1, since if goal is reached chance of success = 1
S₀ = 0, since if gambler has 0 units chance of success = 0
S_bank = win*S_bank+1 + loss*S_bank-1 + tie*S_bank

Since win + loss + tie = 1

1. (win + loss + tie) * S_bank = win*S_bank+1 + loss*S_bank-1 + tie*S_bank
2. win*S_bank + loss*S_bank + tie*S_bank = win*S_bank+1 + loss*S_bank-1 + tie*S_bank
3. win*S_bank + loss*S_bank = win*S_bank+1 + loss*S_bank-1
4. win*(S_bank+1 - S_bank) = loss*(S_bank - S_bank-1)
5. S_bank+1 - S_bank = loss/win*(S_bank - S_bank-1)
6. if bank = 1, S₂ - S₁ = loss/win*(S₁ - S₀)
7. since S₀ = 0, S₂ - S₁ = loss/win*(S₁)
8. if bank = 2, S₃ - S₂ = loss/win*(S₂ - S₁)
9. since S₂ - S₁ = loss/win*(S₁), S₃ - S₂ = (loss/win)² * S₁
10. in general if bank = x, S_x+1 - S_x = (loss/win)^x * S₁ where (0 < x < goal)
11. Sum(S_x+1 - S_x) = Sum((loss/win)^x * S₁) as x varies from 1 to x
12. Sum(S_x+1 - S_x) = (S₂-S₁)+(S₃-S₂)+(S₄-S₃)+....+(S_x-S_x-1)+(S_x+1-S_x) as x varies from 1 to x
13. Sum(S_x+1 - S_x) = S_x+1 - S₁ = Sum((loss/win)^x * S₁) as x varies from 1 to x
14. S_x+1 = S₁ + Sum((loss/win)^x * S₁) as x varies from 1 to x
15. S_x+1 = S₁ * (1 + Sum((loss/win)^x) as x varies from 1 to x
16. since (loss/win)⁰ = 1, S_x+1 = Sum((loss/win)^x) as x varies from 0 to x
17. Sum((loss / win)^x) is the sum of a geometric sequence whose initial term equals 1
18. S_x+1 = S₁ * (1 - (loss/win)^x+1) / (1 - loss/win)
19. Let x = goal - 1 and since S_goal = 1, 1 = S₁ * (1 - (loss/win)^goal) / (1 - loss/win)
20. S₁ = (1 - loss/win) / (1 - (loss/win)^goal)
21. substituting 20 into 18, S_x+1 = (1 - (loss/win)^x+1) / (1 - (loss/win)^goal)
22. if x = bank - 1, S_bank = (1 - (loss/win)^bank) / (1 - (loss/win)^goal)
23. S_bank = (1 - (loss/win)^bank) / (1 - (loss/win)^goal) = probability of reaching goal with bank units

Let R = probability of ruin
R = (1 - probability of success) = (1 - S_bank)
R = 1 - (1 - (loss/win)^bank) / (1 - (loss/win)^goal)
R = (1 - (loss/win)^goal) / (1 - (loss/win)^goal) - (1 - (loss/win)^bank) / (1 - (loss/win)^goal)
R = ((loss/win)^bank - (loss/win)^goal) / (1 - (loss/win)^goal) = probability of of being ruined with a bank of bank units and a goal of goal units