ultrafilter, I’m surprised at you. 
If you filp a fair coin a lot the expected value of the difference between heads and tails increases and is equal to the square root of N, with N = number of tosses.
The probability of getting exactly 50% heads is zero whenever N is odd and increases to 1 as N approaches zero. The probability when N is even is
C(N, N/2)/2[sup]N[/sup].
This value approaches zero with increasing N.
FWIW, I’ve seen a flipped nickel end up on its edge. It’s uncommon, but with a relatively thick-edged coin like a nickel, a smart bounce off of a tabletop can cause the coin to end up on edge.
I wasn’t clear. I was speaking of the relative difference, which has an expected value of 1/sqrt(N). That decreases to 0. My bad.
Neither do I. 
Well before all these great explanations, I would have said, The chances of flipping 1,001 heads must be astronomical, it’s “due” to flip tails.
Of course I understand why this is wrong now.
Princhester, everyone else too, great posts.
(Someone check my math and logic here)
If I had $127,000 to use and wanted to make a quick $1000 would I have a 93% chance of doing it by betting black on roulette?
Chances that roulette wheel will come up red 7 of 7 times are 1 in 14 or only 7% (disregarding 0 and 00).
If I win on any spin I immediately stop and walk away with $1k.
1st spin bet - $1K
2nd spin bet - $2k
3rd spin bet - $4k
4th spin bet - $8k
5th spin bet - $16k
6th spin bet - $32K
7th spin bet - $64K
Even with your 93% edge the house still has that 7% chance to take your $127K.
Does it sound worth while?
What are the odds with 0 and 00?
Hampshire, the strategy which you describe is called a Martingale. Using that strategy, you do in fact have a large chance of winning a small amount, and a small chance of losing a large amount. Your expectation is still exactly the same; for every dollar you put on the table you’ll get back, on average, 90-some cents. Nonetheless, it’s still probably a bad betting scheme (worse than, say, just putting $1000 on each spin), because you’re guaranteeing that you won’t get a big, exciting payoff (which is probably why you’re at the roulette wheel in the first place), you’re cutting short the amount of time you spend playing (you’re having fun, right?), and you’re leaving a very real possibility of losing all of your gambling money in a hurry (and then what fun would your Vegas vacation be?).
Worth noting, I hope, is that no betting scheme can improve your expectation. So if you use the Martingale or the Anti-Martingale or some other way of responding with your bet size to wins & losses, you still expect to walk away with a certain amount of money. What these schemes can do is change the odds of some particular outcome such as: How do I maximize my odds of taking as long as possible to go broke? How do I maximize my odds of winning a certain % in a given time? How do I minimize my odds of losing a certain % in a given time?
These may be worth knowing for your Vegas Vacation—i.e. going as long as possible w/out going broke; however, no scheme will help you make a career out of gambling unless by “career” you mean becoming a casino owner. The only reliable book I know of that discusses this is Epstein’s The Theory of Gambling and Statistical Logic. It is a heavy book; but the fact that you find it in the math section and not right next to How to Win at Slots and Beat the One-Armed Bandit by “Diamond” Bugs Murphy and Leftie McFee, respectively, says a lot IMO.
No. The context of the “gambler’s fallacy” is a “fair coin”, which is by definition evenly weighted and flipped with equal chances for heads and tails. Even stepping outside this to “pattern finding”, only moves us from probability (loosely, predicting chances for events from theoretical considerations) to statistics (determining chances for future events by analyzing past events).
So: if you’ve got a “fair coin”, the math is always right. If you’ve got a real coin and are incorrectly assuming it behaves like a fair coin, of course the predictions from probability will be wrong. If you throw it a bunch of times and use statistics, you can find an idea of the real probability it lands heads or tails (and a feeling of how accurate your idea is). Even so: now you’ve got an event with some probability P, and separate tosses are independant, which brings us back to saying that the gamblers’ fallacy is wrong.
There is nothing “pushing” a sequence of independant Bernoulli trials to succeed in proportion to the probability of success of each individual one, it’s a convergence result that is more and more likely to hold as the sequence gets longer without end.
Two quick, unrelated observations:
If you are absolutely forced into making a bet, you are betting off picking a “hot” number than a “due” number. There’s always a slight chance that the game is rigged, either intentionally or accidentally, so you might as well capitalize on it. If you’re wrong, you’re in no worse a position than if you had chose your bet randomly.
The British roulette guy was monumentally stupid. Even discounting the 0 and 00 on the roulette wheel, his odds were far from 50/50. He has to pay at least 35% taxes on his winnings. Since he’s a British national who won his money in America, he might have some complicated international income tax issues that will cause a major headache and incur some legal bills. (But given that he has no other income or possessions, it probably won’t be that complicated.) And, virtually everything he sold had decreased significantly in value. He may have gotten only a few hundred bucks for his entire wardrobe, which will cost in the thousands to buy back.
In re: the roulette guy. Have we determined that this was a real anecdote and not a misremembering of Kipling’s “If–”?
You mean this guy?
As to the Gambler’s Fallacy, I think it was best expressed by Clive Owen’s character in Croupier:
Minor correction: actually the probability of 7 reds in a row is (18/38)[sup]7[/sup] — about 0.535%, or 1 in 187. More relevant to your strategy though, the chances of 7 non-black numbers in a row is (20/38)[sup]7[/sup] — about 1.12% or 1 in 89.
This doesn’t change your strategy’s expected rate of return however, which is still positive in the casino’s favor.
Unless the scammer is waiting for a hapless punter to take advantage of. Then all bets are off, so to speak.
Gambling and insurance are two sides of the same coin. They both trade in risk—casinos sell it to you and insurance companies buy it from you. It is perfectly reasonable to say that one enjoys playing the slots, even though one is virtually guaranteed to suffer a net financial loss from it over one’s lifetime, if one enjoys the risk of plunking nickels into the machine and hoping for a payout. That is a fair market exchange that one is free to make.
The Gambler’s Fallacy is about not understanding the underlying concepts.
Which is not to say that many of those who gamble aren’t described by the quote you provided. I think I’m probably just nit-picking at this point and should shut the hell up.
Great, just when I start to understand, you guys take it to the next level and start talking about Wink Martindale or something.
What’s the story with 1st spin 1k, 2nd spin 2k, etc?
I should have specified that the quote refered to compulsive gamblers (Owen’s character is a recovering one). I think most casual gamblers (myself included) realize that they will lose in the long run and only play for fun.
[QUOTE=Phnord Prephect]
Please, allow me to throw a nice big wrench into the works of this otherwise logical and correct discussion:
[QUOTE]
Heh. I figured this out a long time ago as a kid. As a result, I usually got to pick where we ate out when my sister and I disagreed. I’d always look at the coin in our Dad’s had right before he tossed it, and quickly call out whatever side wasn’t showing (since he’d turn it over when he caught it). Very often the coin wouldn’t actually flip at all in the air. I wouldn’t be at all surprised if, when it did flip, it tended to flip a consistent number of times, just as a piece of buttered toast will typically flip approxamately .5 times in the distance from the kitchen counter to the floor. (Yes it was a typical, modern, kid-centric household, and my sister and I got to decide where we ate. Neither of us is in jail yet, and I at least have very catholic tastes for food.)
BTW, great stories, Jurph. An excellent way of refuting the Gambler’s Fallacy. Did you make those up, or did you steal them from someone? (John Allen Paulos, maybe?) Either way, I plan to steal them from you if I ever have a discussion of the Gambler’s Fallacy.
someone please disabuse me of this notion:
using Hampshire’s method and bytegeist’s calculation, a 10 non-black streak would be about 0.16% or 1 in 625. so if i would to say, wait for 3 reds to appear then start betting, does that mean i have about 1 in 625 tries to lose the 127k? does that mean i can win 625k during which i will lose only 127k??
i know that the expectation remains the same and varying the bets does not change the odds. this is especially clear when you are on that 64k bet - you still have a 10/19 chance of losing it.
so what is wrong about the notion exactly?
Well, I’m not a math major, but until one comes along, let me try.
If, at the beginning of 10 trials, you say, “What are the odds that the next 10 will not come up black?” the answer is 1 in 625. That’s correct.
If, at the beginning of seven trials, you say, “What are the odds that the next 7 will not come up black?” the answer is 1 in 89. That’s correct.
In other words, it doesn’t matter what happened in the past. If you jump in after any possible combination of any number of trials, the odds that the next 7 numbers will be non-black is 1 in 89. Always (assuming a fair wheel etc…)
Look at it jk1245’s way:
Chance of first three (that have already happened) being non-black=1
Chance of next seven being non-black=.0112
Chance of first three (that have already happened) being non-black AND next seven being non-black= 1*.0112=.0112
i think i understand what you’re trying to say, but why can’t i walk in with 127k betting that a streak of 10 will happen only 1 in 625 tries, so i can slowly win 1k by 1k in the meantime?
If you walk in with 127K and use a Martingale scheme, then you probably will win a grand. But you might lose 127 grand, without winning anything. The first possibility is more likely, but it is not 127 times more likely, so the Martingale scheme (like every other gambling scheme) is a loser.