Gambler’s Ruin


I discussed random variables,discrete probability distributions and expected values in my data analysis class this week. After the mundane definitions and examples, I threw at them, a variant of the gamblers ruin idea and asked them if they would be willing to bet their money on it. The idea goes like this:

I have a coin toss game where I give you 2 times your bet if you win and nothing if you lose. Assume we have a fair coin, would you play the game with me and bet your money? If you will, then what is your strategy, assuming you are in it to win…

It turns out that a few in the class wanted to play the game, but did not know the best strategy to win. These were risk takers and willing to bet their money with a 50-50 chance. Much of the class was conservative and didn’t want to play.

There were a handful who think gambling is bad!!.

There is a winning strategy in this game and if you have enough money to bet, there is a high chance that you will double your initial bet. I start with a bet of $x. If I win in the first toss, I will take my $2x and leave. If I lose in the first toss, I will double my bet to $2x and play again. If I win, I get $4x. Since my investment was $x + $2x, I make an additional $x. You can keep doing this till you win, but the first time you win, you should stop.

The game follows a geometric distribution, with a Pr(X = k) = (1-p)^{k-1}*p, probability of first win on the kth try. Since winning probability is 0.5 for a fair coin, the chance of winning it in the first try is 0.5; the chance of winning it in the second try is (0.5)(0.5), the chance of winning in the third try is (0.5)(0.5)(0.5) and so on…So, you will have a decent chance of winning (more than 90% chance) within the first 5 trails. Just stop playing it the first time you win..and take your money.

I know, it is easy to throw numbers and show that I can win… but will I bet my bucks on it? I can perhaps bet a $100 and win back my $100 plus an additional $100, because i can rise the bets up to a limit.. I am not willing to bet $1000 or more …

Here are some gambler’s ruin ideas.


  1. If I understand this theory correctly, you are saying that since the probability of coin toss resulting in gamblers favor is 0.5, the winning chance is multiplied by 0.5 every time a gambler places a bet. By the same theory, isn’t the chance of losing also multiplied by 0.5 times every time a bet is placed?

    1. The chance of winning and loosing is 50%, but if you double your bet each time you loose up to to the first time you win, you will make up for the loss and double your initial bet. The key is to stop playing the first time you win.

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