# Thread: The Gamblers Fallacy and Regression to the Mean

1. ## The Gamblers Fallacy and Regression to the Mean

Hi, so first time posting and all.

I'm not a statistician by degree or trade (actually an engineer), so a little confused by something, and was hoping someone might be able to help.

The confusion surrounds two ideas, The Gamblers Fallacy and Regression to the Mean.

As far as I understand them, the Gamblers Fallacy is the idea, say for a coin toss, getting 5 tails in a row makes it more likely to get a heads (so you gamble more - reds and blacks on a roulette table is another example). I think I've been told this is wrong because it assumes data points are dependant of each other, whereas they're actually indepandant (so whether you get heads or tails on the previous toss, the next toss is still a 50/50 chance).

Regression to the Mean on the other hand, (quoting from wiki here) 'is the phenomenon that if a variable is extreme on its first measurement, it will tend to be closer to the average on a second measurement'. A common example I've heard used is a batter in baseball - if they're hitting above average, it's not so much that they're having a particularly amazing game, they're just having a slight stats fluke, and soon they'll regress to the mean.

My confusion (and very possibly yours after reading that garbled mess) is why, when you're gambling do you treat the data points as indipendant, and not gamble on the coin toss averaging out over time (or regressing to the mean), but when you're looking at regressing to the mean do you seem to treat the data points as dependant as assume the 'hot' streak in batting will average out.

If you understand any of that mess, kudos - like I said, not a statistician, so the terminology is a bit off. If you have any ideas even better.

Thanks for any help.

2. ## Re: The Gamblers Fallacy and Regression to the Mean

Originally Posted by Mr Ben
My confusion (and very possibly yours after reading that garbled mess) is why, when you're gambling do you treat the data points as indipendant, and not gamble on the coin toss averaging out over time (or regressing to the mean), but when you're looking at regressing to the mean do you seem to treat the data points as dependant as assume the 'hot' streak in batting will average out.

For the gambler's fallacy, many people believe that subsequent events are related, when in reality they are not. Take a die. A person who rolls 10 times and does not get a 6 might think that getting a 6 is more likely b/c of the 10 previous rolls. While rolling rolling 11 times and not getting a 6 for all 11 rolls is unlikely, your chances of rolling a 6 given that you've rolled 10 times is still 1/6 (the same as if you only rolled once). In the first scenario, you're considering an overall probability whereas in the second scenario, you're considering a conditional probability.

For regression towards the mean, it's better if we approach this from a frequentist's POV. Frequentist statisticians (also known as classical statisticians) believe that in a process where the variables observed are iid (independent and identically distributed), there are fixed parameters that can be estimated. The mean is one such parameter. This fixed value doesn't change. So connecting this reasoning to your example, the batter has a (unknown) "true" batting average which we can try to estimate. Hypothetically, lets say that we do know what his "true" batting average is and that each time he goes up to bat is independent. Then if one game, he has a killer day and scores home runs on every bat, you'll have observed an event that is way above his true batting average. The probability of this is extremely rare and future observations will likely be lower (i.e. closer to the mean). You're not in any way assuming that the observations are dependent. You're just observing a value that is rare. Values that are closer to the mean are more likely to happen and therefore the next observation you see will be closer to the mean.

Hope that helped.

3. ## Re: The Gamblers Fallacy and Regression to the Mean

I am drawn by "Mr Ben's" logic. At what point do you take heed to an abnormal roulette streak that should expierience some regression to the mean. I'm sure an Ivy Leaguer would argue that there is no point because each trial is an individual event. But still, the documented world record for consecutive spins on a roulette wheel is 31 straight red. (So the whole infinitive arguement isn't valid on trials where mean is 50/50.) What would impress you??? 20 in a row, 25 in a row???

4. ## Re: The Gamblers Fallacy and Regression to the Mean

Originally Posted by DHass22
But still, the documented world record for consecutive spins on a roulette wheel is 31 straight red. (So the whole infinitive arguement isn't valid on trials where mean is 50/50.) What would impress you??? 20 in a row, 25 in a row???
i think i'm missing your point here. if the roulette wheel is fair then there is a probability (albeit small) that there could be 31, 32, 33, 34 straight reds.

you know what would impress me? if in the whole history of the universe from the inception of time, after spinning a (fair) roulette wheel endless and endless of times, you NEVER get 31 (or 32, or 33, or 34, or pick your number regardless of how ridiculously high it is) straight reds.

it is akin of the (incorrect) reasoning that people apply to unusual events and label them 'miracles'. like "what are the chances that such and such person was spared from this or that horrific accident" or "what are the odds that i was thinking about this and that and it just happened"... well, the probability is clearly not 0 so if 'miracles' (or any other unusual event) never happened...well... now *that* would be a miracle

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hlsmith (10-10-2013)

6. ## Re: The Gamblers Fallacy and Regression to the Mean

OK - A practical question from my untrained perspective.....two football (college or pro) are scheduled to play in the upcoming weekend game. Team A has covered the point spread (let's say for example) 4 times in a row. Team B has failed to cover the point spread 4 times in a row.....I have read that the probability team A will "win" this game against the point spread in such a situation is very very small. Is this accurate? Or is my query even framed in such a way that it can be answered by trained statisticians......any thoughts will be appreciated.

7. ## Re: The Gamblers Fallacy and Regression to the Mean

Bookmakers try to ensure that the odds of team A winning is 50%, but most of all, they want to ensure that an equal amount is bet on A and on B. They study many statistics, I don't know how much they look at record against the spread.

If it were true that the probability A will win is very, very small, everybody would bet on B. Then the bookmaker would lower the spread to get more people to bet on A.

There are all kinds of "rules" like a West Coast team traveling east, a given coach's record after the bye week, a quarterback's record when the temperature is below 40 degrees, but the bookies know these rules and adjust the spread accordingly.

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