# Long-term chances of winning/losing in betting systems

Okay, this has been bothering my quite some time now, it might be difficult for me to explain it but please bear with me:

I will be using Martingale negative progression betting system for this example. For those of you who don't know what that is, you don't have to know the specifics but it's a betting system where you keep making steady profit until you encounter a very unlikely event (let's say 1% chance) and you lose all your money. The method is said to be a good "short-term" strategy as the chances of you encountering that event are very slim, but the longer you play the higher the chances of you encountering one. So just for the sake of it, let's say your chances of making profit is 80%, 19% chance of losing some money and 1% chance of losing ALL money per 100 games (thus "a good short term betting strategy"). As the number of games increases, your chances of profiting go down, eventually converging to 0% after infinite amount of games.

Now this is the part that I can't wrap my head around. If you take a LARGE number of games and split them into multiple "short term" sessions with 100 games per session, WHY doesn't that reset your chances back to 80%?

I know the chances are decreasing and odds of you losing are accumulating within each extra game played, but in what manner? Does the decrease follow some kind of pattern or a function (e.g linear decrease)? How does one predict or at least get the idea of when the "short term profitable strategy" starts to become a "long term non-profitable strategy"? Is there such thing as an optimal "mid term" solution, or is it just a concept of my imagination?

My current theory is that each individual betting system follows a specific pattern of your chances decreasing within number of games played, even though it might look like a gold mine in short term. That pattern can be simulated mathematically but given the nature of probability itself, it would be very hard to predict WHEN that strategy is starting to bring you net profits of less than 0. What do you guys think?

I would really appreciate any insights or suggestions you have regarding this topic as it's been bugging me for a while now!

• the chance of surviving $n$ games is $0.99^n$ which goes to $0$ for large $n$. But I think you have a conceptual misunderstanding of short/long term profits. If the trials are independent, there is no way to make a long-term losing strategy into a short-term winning one. – Slug Pue Feb 10 '17 at 17:41
• @SlugPue depends on how you define these ideas of "long-term losing" and "short-term winning"..for example, many typical gambler's ruin problems have positive expected value, but probability of ruin going to 1 if you play forever. – David Feb 10 '17 at 17:43
• @David sure, but for any realistic situation, i.e. with almost surely bounded lifetime and finite wealth, you can apply optional stopping theorem and show that it never works. – Slug Pue Feb 10 '17 at 17:49
• But what about an optimal solution? We define "long term" as something realistic, let's say 1000-10000 games. Then if you still have the edge over those first 1000-10000 games, why wouldn't the system profit you? – Shibalicious Feb 10 '17 at 19:36

Now, I'm not sure that you've described a typical Martingale negative progression betting system, but going by your definition, it appears that the answer is simply, once you lose all your money you stop playing. That is, it doesn't matter how much money you have, once you lose, you're done. Therefore, because the unlikely event will happen to you eventually, you can't possibly end up with more than $0$ dollars if you play forever.

So you can't split the sessions into 100 games, because if you played them in sequence, you would start them with different wealth. Eventually, you would start one of those 100 game stretches with $0$ wealth (and indeed, every 100 game stretch after you've lost all your money). And so you're odds of winning are still 80% on that stretch of games, but you have no money to play with.

• But what if I start every new "short term" session with the same initial balance every time? Let's say first 1000 games you have an edge over the house. Wouldn't that mean that it's actually profitable for you to play in that amount of games? And if so, how do you know know when your chances have stopped being in your favour? – Shibalicious Feb 10 '17 at 19:39
• I think it's very difficult to explain things better without a well-defined game, betting, and payout structure. It sounds like you aren't describing something that is possible, but I can't be sure because you haven't defined your problem well. – David Feb 10 '17 at 20:36

Breaking up your sequence of games into separate $100$-game sessions can be a good idea, but only because the alternative is a remarkably bad idea.

If you start each new session with the same bankroll (putting any prior winnings in savings, and taking money out of savings if your last session was a loss), then you can consider each $100$-game session as a kind of "super-game" that has multiple possible win/loss outcomes rather than just one possible "win" and one possible "loss." Indeed, in that case your modification of the system actually does "reset" the probability of profit to $80\%$ at the start of each session.

The fly in the ointment is, that's only an $80\%$ chance for that session, not for that session and all others up to that point. Moreover, unless you actually have an edge in each individual game (that is, unless you are a casino owner or someone with a similar advantage), the $80\%$ will be an $80\%$ chance of winning a relatively small fraction of your initial bankroll, whereas the $1\%$ is a $1\%$ chance of losing the whole bankroll. So this is a "game" in which you accumulate small winnings over many consecutive winning sessions, only to have them more or less wiped out by the occasional large loss.

The long-term results will be that the total of your available assets (the bankroll for that session, plus any amount you have saved up from past sessions) will make a uniform random walk over its possible values, and it can be analyzed that way. If you set some sort of limit after which you never play again--after $N$ sessions, or after your total accumulated winnings are $M$--then you have some probability of walking away with a net profit. How much probability depends on how modest your goals are. If you do not set some sort of limit, and the game is not unfair in your favor, then eventually you will be wiped out by this strategy too, though probably not as quickly as if you simply continued the Martingale.

It is not necessarily instinctive to catch this one but one specific thing ties all this together. The situation assumes that you always need to end your session with a win (which would compensate for all your losses). So the theory assumes that you don't stop your session if you are losing and you keep going until you win.

That's where the hammer hits. Shorts sessions are good but you can't 'decide' to make it short with this theory because you need to keep going until you win. If you would start each session with a fixed amount you could lose all your money and it might not even out for the other sessions where you will have small gains. This theory wants to give you high chances of winning but low returns while risking the totality of your money.

For example, by trying to make gains of 5\$, you may end up betting 160\$ if you 5 times in a row. If you would end up losing that amount during that one short sessions, you would have to win 32 times in a row without losing to get that amount back in the next session (which makes it really hard to come back from).

• Hi Leucippus, i thought that the main question was paragraph 3 "Now this is the part that I can't wrap my head around. If you take a LARGE number of games and split them into multiple "short term" sessions with 100 games per session, WHY doesn't that reset your chances back to 80%?". My answer was that "short" sessions are not profitable if they are stopped "short" and that if you stop it on a losing streak you are losing too. Maybe the "asker" had a few questions bundled in one and I was aiming at answering all of them with this answer. Should I edit my answer or delete it? – Dan Aug 23 '18 at 3:11