# Gambling Math: Probability of Bankruptcy Over N Bets

• h0dgey84bc
In summary, the conversation discusses the calculation of the probability of going bankrupt over a certain number of bets in a bookmaker with a $1000 balance and bets of$500 with a 0.5 probability of winning. The discussion includes examples for 2, 3, and 4 bets, as well as the question of generalizing the calculation for any number of bets. It is also mentioned that the game would only end when the player goes bankrupt, and the possibility of stopping the game and withdrawing after winning a certain amount of money.
h0dgey84bc
Hi,

Let's say I have $1000 in a bookmaker, and this bookmaker would only allow bets of$500 at a time, each betting having a probability of 0.5 to win (this means odds of 1:1 or 2.0, so if you win you win 500, if you lose you lose 500 staked). How do I calculate the probability of going bankrupt over N bets, i.e. P(N)?

I know when number of bets is 2, the outcomes are:

WW (+500+500. leaving 2000 balance) 25%
LW,WL =>balance of 1000 still, 50%
LL=> balance of 0, bankrupt, 25%

then for 3 bets, the outcomes are(remembering we would of stopped playing if LL had happened and bankrupt us after 2 bets)

WWW(2500)
LWW(1500), WWL(1500),WLW(1500)
LWL(500),WLL(500)

so no chance of going bust here except if we had already done it after 2 bets, so still 25%.

After 4 bets we could have (excluding the bets we went bust after 2 times)

L=0::WWWW(3000)(6.25%)
L=1::WLWW(2000),WWLW(2000),LWWW(2000),WWWL(2000)(prob is 25%) (4!/3!1!=4 combos with L equals 1, and therefore balance of 2k)
L=2::LWLW(1000),WLLW(1000),WLWL(1000),WWLL(1000),LWWL(1000)(prob is 31.25%) (4!/2!2!=6 combos with L=2, but one is LLWW, which is bust after two so excluded)
L=3::LWLL(0),WLLL(0) (prob: 2*(0.5^4)=12.5%...4!/3!=4 with L=3 , but two are LLLW,LLWL,which are excluded as they bust after two)
(the other 25% is for times we went bust on first two, i.e LL...)

Therefore the TOTAL prob of going bust after 4 moves is P(4)=P(2)+12.5%=37.5%

How do you generalise this to get the probability of busting for any number of bets N?

Is the game allowed to finish, even if you're accumulating debts?

well you would only stop betting when you went bankrupt, there's no way to build debt, if you keep winning you're making profit on the original 1k. If it makes the problem simpler though I'd still be interested in seeing the answer, when you stop playing and withdraw when you'd won say 10k(not sure if that would actually make it simpler though, since still infinite paths)

This problem seems like it should be simple, but I can't seem to get the answer for the life of me, haha, feel like it should converge to 100% as N->infinity...

Last edited:

## 1. What is the purpose of studying the probability of bankruptcy in gambling math?

The purpose of studying the probability of bankruptcy in gambling math is to understand the likelihood of losing all of one's funds while participating in a series of bets. This information can help gamblers make informed decisions and manage their bankroll more effectively.

## 2. How is the probability of bankruptcy calculated in gambling math?

The probability of bankruptcy is calculated by taking into account the initial bankroll, the size of each bet, and the expected win rate of the game being played. This calculation is based on mathematical principles and can vary depending on the specific game and betting strategy.

## 3. Can the probability of bankruptcy be reduced in gambling?

While the probability of bankruptcy cannot be completely eliminated in gambling, it can be reduced by following certain strategies such as setting a limit on the number of bets or the amount of money to be wagered, choosing games with lower house edge, and practicing responsible gambling habits.

## 4. Are there any risks associated with relying on the probability of bankruptcy in gambling?

Yes, there are risks associated with relying solely on the probability of bankruptcy in gambling. This calculation is based on mathematical probabilities and does not take into account external factors such as luck or human error. It is important to also consider personal limits, emotions, and other factors while making gambling decisions.

## 5. How can understanding the probability of bankruptcy benefit gamblers?

Understanding the probability of bankruptcy can benefit gamblers by helping them make more informed and responsible decisions. It can also help them manage their bankroll more effectively and reduce the risk of losing all of their funds in a gambling session. Additionally, understanding this concept can also help gamblers identify potentially problematic gambling behaviors and seek help if needed.

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