"Gamblers Ruin" and bet size question ????

[LucaBrasi]

Here is the assumption. You pick 55% winners.
I know, a big assumption, but stay with me.


If you had a box with 550 white marbles and 450 black marbles, each representing a winner and a loser, what would be the longest consecutive streak of black marbles (losers) you would draw?

I ask this as a way to correctly gauge an appropriate bet size for a given winning %, and to avoid "gamblers ruin" with too large of a bet.

Is there a mathematical formulae that would work if the above numbers were changed, like the win %, the # of trials, etc.

All math whizzes, your responses are appreciated.

 

[DarrylParsons]

The so-called Kelly Criterion says your optimal strategy is bet the percentage of your bankroll which corresponds to your % advantage.

In this scenario you have a 10% advantage (make 100 picks for $1 each and you will have $110 at the end if you win 55 times) so you should bet 10% of your bankroll. If you lose, then don't forget that your next bet will be 10% of your NEW bankroll (which just got smaller because you lost)

But if you think you have a 10% edge in sports betting, you are kidding yourself. Assume 1% and you should be fine.

 

[LucaBrasi]

From everything I've read, the Kelly strategy is suicide.

I limit myself to 1-2% per play personally, but I was interested if anyone had a knowledge of probability, one that would say, "in this number of trials, with this % of success, there is a streak of 9, 11, 20, "x - ", whatever losers out there waiting for you.

 

[JamesR]

Someone else can present a math equation, my math skills peaked when I was 16. A major league baseball team at 55% would go 89-73 for the season. It would be interesting to look back at some team logs, but it wouldn't be odd for a team with a 89-73 record to have a six or seven game losing streak over the course of a season. I would also think such a team would have a 3-12 or 4-11 run over 15 games as well. Since you're increasing the sample size the chance for a more extreme stretch of "losing" or picking the white marble would be higher. My guess for the average if you actually did such an exercise would be nine or ten.

 

[Bucky]

Let's assume that Mr. A could lose 10 in a row. That's all well and good but he could also lose 9 of the next 10. It's like flipping a coin - after 10 consecutive heads the odds are still 50-50 on the next throw. So, I don't think the percentile chances of 10 losses in a row means anything as nobody gambles 10 things and quits. What happens over a 1000 occurrences is more important. 2% works for me, also.

 

[drunkguy]

chance of two events occurring is just the product of their individual probabilities...

Think about it like a set of dice...chance of rolling a 6 is 1/6. Chance of rolling two dice and having them both be 6 is 1/36 (1/6 times 1/6).

If you are a 55% capper, your chance of losing one game is 45% or .45. Chance of losing two in a row is .45 X .45 = .2025. Chance of hitting both is .55 X .55 = 3025. Chance of going 1-1 is the rest, .4950. Here is it broken down for the number of consecutive losing wagers in a row:

1 game = 45%
2 games = 20.25%
3 games = 9.11%
4 games = 4.10%
5 games = 1.85%
6 games = 0.83%
7 games = 0.37%
8 games = 0.17%
9 games = 0.08%

In other words, there is less than a 1 in 1000 chance of you losing 10 in a row. It looks like this kind of math is what you want to be able to do. But as bucky mentioned losing 9 out of ten would be more more common and something else you have to think about..you also have to realize that estimating your win% is difficult and the numbers are going to change if you are closer to a 50% capper.

As far as bet sizing, I wouldn't recommend going above 4-5% of BR on the strongest plays. Ideally the majority of your bets will be 1-2% of BR.

 

[loser]

Typically the "gambler's ruin" question looks at what size bankroll is needed to double your money versus going broke for a certain % advantage.

If you can pick 55% winners your advantage is roughly 5% after incorporating the vigorish. For example, if you make 100 bets at $11 each for a total of $1100 you will win 55*10 = $550 and lose 45*11 = $495, for a profit of $55, or 55/110 = 0.05 (5%).

If you have a 5% edge and are betting 2% of your bankroll each time then you have divided your bankroll into 50 units. The gambler's ruin formula shows that your probability of doubling your money begore going broke is 0.9933. Thus, this is very conservative.

If you use the Kelly Criterion you would bet 5% of your money on each game. This criterion is just one particualar risk/reward utility function, but a popular one. This would give you 100/5 = 20 units, and your probability of doubling your bankroll would now be 0.88, or about 88%.

Your question about how many bets you might lose in a row is actually a more complex one. For example, with a 5% advantage you have a 0.525 probablity of winning a game and a 0.475 probability of losing a game sine 0.525-0.475 = 0.05. Suppose you were to bet 8 ganes on a given weekend. Your probability of losing all 8 games would be 0.475 to the 8th power, or 0.00259. Quite small. However, if you do this for a season (say 16 weeks) your probability of losing all on one of the weekends is about 4%. If you are an old timer (like myself) and have done this for say 32 seasons, your probability of losing all 8 games at least once is now roughly 73.5%. The point being that if you bet enough you will see sizable losing streaks. However, the relevant question usually is how much of a bankroll you need to comfortably weather those streaks.

Keep in mind that this assumed a 5% edge. With a smaller advantage the probabilities change considerably.

Hope this helps. If not, ignore it.

 

[WildBill]

It would be ridiculous to try to tie in gambler's ruin with how many games you lose in a row. Think of it this way, if your bankroll is 20 units and you never go more than 9 losses in a row, are you safe? Of course not. What if you go 4-24? I am guessing this question was just asked for Martingale purposes. I won't even go into that other than to say DON'T DO IT.

A lot of people will talk about the Kelly criterion, but it is quite difficult to work with sports. The fluctuation of your edge makes it very inefficient. Hence a lot of people do things like 1/2 Kelly or 1/3 Kelly. It assumes that while you might on average have x% ability, chances are some of your picks are just inherently going to be of lower quality so you must cut back that optimal bet percentage for a number of reasons.

To keep it simple, the Kelly system and numerous other methods could take up a 200 page book so we aren't going to resolve a lot here. I always make the suggestion that bet sizes be kept flat and that you start out your BR with 30 units, if you aren't using a terribly high amount for your starting BR. If you really would be hurt losing that BR and would be relegated to betting far smaller stakes if you lost it then get it up to at least 50 units. The peace of mind is worth almost as much as the extra cushion. Going forward don't raise your bets until your BR has gone up enough to still have 30 or 50 units AND also that you had sufficient games to justify it. If you go say 12-0 and now your bankroll rules say go up, don't do it. Make sure that you have had a sustainable run before moving up or else the downside of that run could hurt you quickly at higher bets. I tend to avoid Kelly for many reasons, but my favorite way to get through to many people is this. In Kelly your smallest bet will be a win, your largest bet will be a loss. You will see many good and bad runs in sports betting, with Kelly your good runs can easily be wiped out by a shorter bad run. Just stick to methodical gains and fairly static bet sizes is my advice.

 

[DarrylParsons]

I agree with what pretty much everyone said, especially with WildBill's postulation that the original poster is looking for a way to justify a Martingale strategy. To pick up on that shows a huge amount of experience in the world of gambling.

But I will have to speak out in defense of the Kelly Criterion because I feel it is being under-appreciated here. When people say things like "full Kelly is suicide, it's better to use 1/2 Kelly or 1/3 Kelly" it shows a lack of appreciation of just how powerful this theoretical result is.

Don't get me wrong, recommending smaller bets is definitely a good thing but it is not because Kelly is somehow flawed. What is flawed is people's perceptions of how large their advantage is.

A typical scenario: A young whippersnapper tries his hand at sports betting, gets lucky in his first year and hits 57% and then proceeds to overestimate his advantage, bet accordingly and get wiped out. I'd be curious how many times WildBill in his long career has seen this happen. I bet it's a large number. It's a rare animal indeed who says, after a 57% year, that probably 51% is pure expectation and the other 6% is luck (which is probably close to the truth).

It's easier to blame something external like the Kelly System than to admit that you may not be as good as you think you are.

Of course it is good enough to follow it only approximately, ie. flat betting until your bankroll gets much bigger or much smaller and then making a quantum adjustment. The difference between that and exact adherence is small, but to say 1/2 Kelly or 1/3 Kelly is just not right. What's right is to say full Kelly with 1/2 or 1/3 the advantage you had previously assumed.

 

[TheGeneral+]

Nice posts guys. Good information.

Thanks

 

[olddog]

Martingale (doubling up after each loss)generally works, but the one time in 10 or 20 that it doesn't work you are dead.

"If you believe that you can't lose seven bets in a row take a walk through the casino and look at the display board that is connected to most roulette wheels showing you the twenty previous spins. How many times do you see one color come up more than seven times in a row? I once saw black appear 17 times in a row! If you started with a five dollar bet and doubled up when you lost, your 17th bet would be $327,680."

And to answer your original question, if you tried the marble routine enough times you would eventually pick over 400 black marbles before a white one showed up, maybe all 450. Very, very unlikely, but possible.

 

[LucaBrasi]

However, the relevant question usually is how much of a bankroll you need to comfortably weather those streaks.

Yes, this is exactly my question. If you were to say, in 1000 bets, there is a losing streak of 15 games out there waiting for you, then a 2% of BR bet would keep you above water.

I do realize that each trial is an independent event, and bears no statistical relation to previous events.

Good info, everyone. Thanks for confirming what I had already supposed.

 

[LucaBrasi]

<BLOCKQUOTE class="ip-ubbcode-quote"><font size="-1">quote:</font><HR>Originally posted by Darryl Parsons:
I agree with what pretty much everyone said, especially with WildBill's postulation that the original poster is looking for a way to justify a Martingale strategy. To pick up on that shows a huge amount of experience in the world of gambling.<HR></BLOCKQUOTE>

Not looking for a way to justify anything, just looking for information from good posters and trying to get past the bitterness. Thanks for your contribution.