The math of gambling | Hacker News Reader

The math of gambling

newscientist.com

18 pointspeter12327 comments

The way she describes winning at roulette is bogus.

First of all, if your money is doubled when you hit the right colour, you don't make any profit in the long run. You will stay on the same amount.

Second of all, there is a 0, and often also a 00 on the roulette wheel, which is neither red nor black. This means that the chance you're correct is less than 50%, so you stand to make a loss in the long run.

There is no way to win at roulette, without using a computer to predict where the ball is going to be, based on the movement and spinning of the wheel. And I seriously doubt that even that is possible using just human observation and a trigger in your foot.

[edit: the author is female, sorry about that]

I think you're misinterpreting the author's point about roulette. She never claimed that you could profit off of roulette in the long run. You can, however, almost certainly profit off of it in the short run. The strategy she describes has one double one's bet if one loses. If you bring enough money to cover 7 losses (which means you would have to have roughly 100 times your initial bet), then you have a >98% chance of coming out ahead. You won't be ahead by much, and the strategy won't work in the long run (i.e. you can only do it a couple of times--when you inevitably end up with 7 consecutive losses it will more than wipe out any earnings you make), but it is nevertheless a legitimate strategy to "beat" roulette in the short term.

I hear what you're both saying. But I wouldn't call beating roulette in the short term a legitimate strategy exactly because it only works in the short run--strategies are supposed to work for longer than that! The only legitimate value I can see is that if you actually enjoy the experience of playing roulette that it's worth paying for it. Anyone who's willing to buy volatility can win in the short term. Otherwise, stick to a game of skill and get some money on the table while you do it. If you're good you can get some vol. and some positive EV.

True, I did misinterpret that. And she does own up to it that after 7 streaks you'd need 1280 pounds if you initially start with 10.

Still, the article states that there's a 50:50 chance of landing on black or red, which is simply not true.

Fair enough. The article also claims that the outcome of a coin flip is 50:50, which is also not true.

You're mixing theoretical odds and practical odds, which might be considered sloppy.

Additionally, unfair coins can trivially be used as fair coins, but unbalanced roulette wheels cannot easily be used as fair roulette wheels.

A fair coin has a roughly 51% chance of landing with the same side up as it started with. See the paper by Diaconis et al. "Dynamical Bias in the Coin Toss."

A fair coin is one on which p = 0.5. A coin that has a 51% chance is clearly not a fair coin. This isn't even what I was talking about, but is still equally easy to turn into a fair coin.

In this case, caller calls in the air after flipper hides the side that starts up. 0.5 * 0.51 + 0.5 * 0.49 = 0.5

For any other case, flip a coin twice. Assign one person HT and the other person TH, reflipping TT and HH. Since p * (1 - p) = (1 - p) * p, the dual flip is a fair flip.

Please don't mention the possibility of sleight of hand next.

Splat, Would you pick out of this hat?

-- There are a hundred tickets in a hat. 99 of the tickets win you $1. The last ticket loses $10,000 dollars.

the sentence "it is nevertheless a legitamte strategy to "beat" roulette in the short term" is absolutely false. The expected value of each roll of the dice is negative, summing a bunch of negative expectations leads to an even large negative expectation.

EDIT: Also, some people will talk about infinite bankrolls, there is no such thing! even if there was, infinity + X = inifinity, why are you gambling?

I would pick out of the hat once, maybe even twice. The point is that I'm never going to sum over a bunch of negative expectations. I beat the probability of losing down to something negligible, play once, and then quit forever.

There is a difference between being willing to gamble and beating a game. It is obvious to see that each time you play this game you are losing $9901 dollars (.99x1+.01x-10000). That is the expected outcome each time you play the game.

Probability is often not intuitive. It is easy to think you "beat" the system because you won a couple bucks.

This game may better illustrate my point.

There is a hat with 100 tickets. 99 of the tickets produce a $1 win. The last ticket requires you kill yourself. Would you play this game once or twice?

I'll agree with your semantics. To me, "beating a game" implies that you can consistently come out ahead over a long period of time. Nevertheless, playing an "unbeatable" game over a short period of time is not necessarily irrational. Although your expected value in your first version of the hat game is -$9901, you are not losing $9901 every time you play. If you play long enough it will average out to be this, but every time you play you will either win $1 or lose $10,000. If the chance of a devastating loss is negligible, it's hardly unwise to make a gamble, even if it has a negative expected value--this is why we occasionally do risky things that have a small chance of killing us, like driving or eating sushi. The problem in thinking in terms of an expected value is that an expected value presupposes many trials--given a single experiment, it's a meaningless concept. Indeed, given a single experiment, both the classical and frequency interpretations of probability are meaningless with regards to what the probability of a given outcome really is. In the case of a single experiment, the only interpretation that makes sense is the Bayesian interpretation.

I would not play your second game, but the reason I wouldn't has nothing to do with expected values (notwithstanding the fact that it's technically impossible to calculate an expected value for an unquantifiable outcome like death). In your first game, I feel that I would come out ahead, and in the second game I feel like it's not worth it.

re: roulette

What the author is describing is a 'Martingale' strategy, and it is indeed a surefire way of making a profit (with one critical caveat: you need an unlimited bankroll).

After each win, you always earn a profit of X (where X is your base bet), regardless of how many rounds you have lost before.

Without a sufficiently large bankroll relative to X, however, probability of bankruptcy is high.

And this is why casinos always have a table minimum AND a table maximum.

No, it isn't. The house can only lose the initial amount.

  I bet 50, and lose.
  I bet 100, and lose.
  I bet 200, and lose.
  I bet 400, and lose.
  I bet 800, and lose
  1600, 3200, 6400, 12800, 25600, 51200, 102400, 204800, 409600, and then...

...I lay 819,200 dollars on the table and WIN! 50 dollars.

The reason they have limits is so that Sheik Abudab can't come in and lay down 500 million dollars and bust them. That is one valid reason to take bad odds: If you can destroy something as a side effect, without really being in danger yourself.

In your Martingale example, the table maximum would most likely be set around $2000. So if you lose 6 times (on your $50, $100, $200, $400, $800, and $1600 bets), you can't double your bet anymore. You'll be stuck with betting $2000 at most, which means you can longer guarantee yourself a profit.

But now that I think about it, you could always just move to another table with a higher limit. Of course, if you're going through all this trouble to make $50, you probably don't have the bankroll to keep going.

Maybe another reason for tables having different limits is that the pit bosses don't trust the dealers (or security?) in the low-roller areas to handle money above a certain amount?

Don't the 'high-roller' tables have a higher minimum bet? Most people don't start out at the $1 table, and then move up to the high-roller table as they keep losing though. Most people just plop down at a table that has a range/minimum that is right up their alley with regards to their average bet. Then they only really move if there is a reason to (like feeling that this table isn't 'lucky' anymore, or the dealer changes, etc).

with one critical caveat: you need an unlimited bankroll

That doesn't sound very practical. And why play roulette for money (rather than just for fun) if you already have unlimited money?

The way he describes winning at roulette is bogus.

New Scientist appears to be in serious decline.

One way that I would 'win' at roulette in Grand Theft Auto: San Andreas was to always bet equal amounts on red/black and even/odd. My reasoning was that there are 4 possible outcomes. I have a 50% chance of breaking even, a 25% chance of winning, and a 25% change of losing.

You probably can't 'win big' with such a strategy, but you've at least minimized the probability of a loss from 50% to 25%. You've also minimized the probability of a win by as much too, though. I'm no stats guru so I don't know how well this would work out on an actual roulette table...

{edit} I should mention that I worked at a casino for a while, and roulette is the game with the highest probability in favor of the house. Blackjack is the lowest. Casinos really only have Blackjack tables because patrons want them, or else they would probably get rid of them pretty quickly. {/edit}

Casinos really only have Blackjack tables because patrons want them, or else they would probably get rid of them pretty quickly.

True, even with a simple strategy you can cut your theoretical loss to about 1-2% (without counting cards). But casinos have blackjack because most players don't know how to play. Most people don't follow that simple strategy (they often have leaflets explaining it too!).

With counting cards you can make it profitable for you. But casinos will make life very difficult for you if they find out you're counting cards. See also the movie about the MIT Blackjack team: http://en.wikipedia.org/wiki/21_(2008_film)

The casino I worked at used some formula to generate the number of 'points' you would earn towards free stuff based partially on the risk of the game that you were playing. They used the ~1-2% number for blackjack (or whatever the low number of when you play what blackjack players call 'the perfect game') rather than a larger number that might be closer to the 'average' skill of players.

This signaled to me that the casinos -- or at least the one I worked at -- in general treat the overall risk of blackjack as the low risk that skilled (non-counting) players can achieve, regardless of the number of people that might come in and ignore the simple, effective strategies to the game. Personally I think that the casinos would rather reduce the risk to themselves from skilled players as well as card counters and just remove all the blackjack tables, sending all those inexperienced gamblers onto games like roulette where they might lose a larger portion of the money they brought with them.

Keno has the worst odds, not that it matters. You're probably better off losing quick and going home sober.

Casinos really only have Blackjack tables because patrons want them, or else they would probably get rid of them pretty quickly.

True, but ... isn't that the case with every game in the casino?

Once upon a time, actually, blackjack wasn't terribly popular; craps was the king of casino games. But after Thorp demonstrated it was beatable there was a surge in demand for blackjack. Turns out, nearly all those new blackjack players weren't really very good at counting and lost money hand over fist. Thorp's work (among others) led to increased casino profits and lent an air of legitimacy to gambling; it wasn't just luck anymore, if you "worked" at it you could profit. [Citation needed].

Not necessarily. Despite having pretty bad odds (in favor of the house) compared to other casino games, many people find roulette to be fun and addictive. The same goes for craps, and really applies to the slot machines.

[Thinking about slots, reminded me that in my previous comment I was talking about table games when I was talking about roulette being the worst odds. I don't know if that's true. I haven't crunched the numbers, but that's what I remember being told as an employee.]

Incidentally, if I wanted to put my math skills to work in a casino, I would play poker. This game is one where the odds have not been stacked against me and a little math drastically alters my expected values.

I never, ever learned anything from a newscientist.com article. Those articles are cute, but one learns nothing deep. I say we go directly to the source! Here is it, Ed Thorp's The Mathematics of Gambling:

http://www.bjmath.com/bjmath/thorp/tog.htm

which is the true "bible" of scientific betting. For the mathematically-inclined, I also recommend:

http://en.wikipedia.org/wiki/Gambling_and_information_theory

http://en.wikipedia.org/wiki/Kelly_criterion

and if you still have some energy left, try this:

A Markov Chain Analysis of Blackjack Strategy http://www.ece.rice.edu/~crozell/courseproj/MCBJ.pdf

Have fun! And remember: life is too short to read crappy articles!

Reply on news.ycombinator.com