Teaching an intelligent guy about a basic statistics concept

[Silent A]

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Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

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This is a good point. I had a very good friend who was well trained in math and statistics (he became a chartered accountant in the end). We were talking about lotteries (powerball, 6/49 type). I mentioned that one could just pick numbers 1, 2, 3, 4, 5, 6. His initial reaction was, "that's stupid, what are the chances of that happening?". To which I answered, "just as likely as any other numbers". He had to think about it for a while before he convinced himself that I was right.

[blah_blah]

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imo the problem is that people don't understand what it means for events to even out in the long run. what the large law of numbers really says is that if you flip fair coins, the number of heads you get in n tosses is n/2 PLUS an error term that goes to infinity slower than sqrt(n).

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I think a simpler explanation is that people naturally want things to "even out" in the short term rather than the long term. "Infinite discrete distributions" is kinduva hard thing to understand.

Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

Now where's my grilled cheese w/ the Virgin Mary on it? I'm hungry.

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obviously a careful explanation of the strong law of large numbers is a bit difficult to give to a layperson, but there's really no getting around it, because it is really what mathematicians mean by 'the long run'.

for example if i flip a coin a million times (a random walk on Z of 10^6 steps), my expected distance from the origin is about 2/pi*1000 ~ 600, even though my expected position is zero (equal heads and equal tails). mathematically, the law of large numbers says that the we get about half heads and half tails + a term that goes to infinity slower than n^(0.5+e) for ANY e>0 (this is related to the central limit theorem). this can still be a long ways from zero if n is large though.

qualitatively the important thing is that the mass of the distribution is clustered around {half tails, half heads} - this is the weak law of large numbers.

[gull]

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Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

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He was talking about using it in a real casino with a real bankroll. Actually, I don't think he had a concept of "bankroll" since he just assumed it couldn't fail.

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Even with an unlimited bankroll and no betting limits, it's -EV.

A sum of -EV bets will never be positive. This is easy to see.

[CrayZee]

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obviously a careful explanation of the strong law of large numbers is a bit difficult to give to a layperson, but there's really no getting around it, because it is really what mathematicians mean by 'the long run'.

for example if i flip a coin a million times (a random walk on Z of 10^6 steps), my expected distance from the origin is about 2/pi*1000 ~ 600, even though my expected position is zero (equal heads and equal tails). mathematically, the law of large numbers says that the we get about half heads and half tails + a term that goes to infinity slower than n^(0.5+e) for ANY e>0 (this is related to the central limit theorem). this can still be a long ways from zero if n is large though.

qualitatively the important thing is that the mass of the distribution is clustered around {half tails, half heads} - this is the weak law of large numbers.

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I guess my thing is that people just have a hard time wrapping their heads around the whole infinity thing. I also prefer the integration of a more cognitive bias perspective/explanation.

Not that I have anything against the law of large numbers type of explanation. But, I mean, imagine if The Theory of Poker resembled The Theory of Gambling and Statistical Logic, all 8 copies would have sold as it's way too advanced for the average gambler. Sklansky's book is hard enough as it is.

I have a hard time seeing the OP's friend understanding such advanced concepts. I'd refer the student to the strong and weak laws of large numbers as an exercise. [img]/images/graemlins/smile.gif[/img]

[SNOWBALL]

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Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?


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No. Martingaling is the same thing as laying odds. Basically if you martingale with 2 bets a 50/50 payout game with vig, then you have a 75% chance to win, but you're getting paid less than a third of your total action, so it's unprofitable. If you go for 3 bets then you have an 87.5% chance to win, but you're getting paid less than laying those odds justify. That's the main thing. With an extremely large bankroll, your odds of winning increase, but still lag behind your money odds, and that's the only relevant thing.

As an aside, why would anyone waste there time making bets at a casino if they have unlimited money?

[Silent A]

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Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

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He was talking about using it in a real casino with a real bankroll. Actually, I don't think he had a concept of "bankroll" since he just assumed it couldn't fail.

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Even with an unlimited bankroll and no betting limits, it's -EV.

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I think this can be debated.

Suppose the rules are as follows: heads gambler wins his bet, and tails he loses it. He can bet any amount at all and has an infinite bankroll. The probability of heads is slightly less than 0.5. The gambler initially bets $1, and doubles this bet every time he gets tails. After flipping heads he returns to a $1 bet.

Now consider a single game to end whenever he flips heads. What are the possible outcomes of a single game?

He has a 100% chance of winning $1 and a 0% chance of losing -$infinity.

Now it's true that if you do the math properly, apply limits etc., that the net EV of each game is -ve, however, the practical outcome of this game is that you will win $1 every time you play it. Which makes this a very strange -EV game.

[ilya]

Try an induction argument with n as the number of the trial.

[ilya]

Ask him to imagine that you're flipping a coin that has Heads on both sides. It comes up Heads ten times in a row. Does he think this coin is now less likely than before to come up Heads on the 11th toss?

[BlueBear]

Point your friend to the "law of averages" and "gamblers fallacy" wikipedia article. I am afraid you are vastly overestimating his or her knowledge of statistics.

[Benholio]

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Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

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This is a good point. I had a very good friend who was well trained in math and statistics (he became a chartered accountant in the end). We were talking about lotteries (powerball, 6/49 type). I mentioned that one could just pick numbers 1, 2, 3, 4, 5, 6. His initial reaction was, "that's stupid, what are the chances of that happening?". To which I answered, "just as likely as any other numbers". He had to think about it for a while before he convinced himself that I was right.

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As it turns out, though, picking 1-2-3-4-5-6 is stupid, since you are punished for picking the same numbers as others.