Regression to the Mean vs. Gambler's Fallacy

How do the two ideas NOT condtradict each other? At what point is it safe to say that an event is more likely to occur because it has not occurred as much as expected over a long enough period of time, even though mathematically the probability of an individual trial remains the same?

I was playing a game of Monopoly the other day, and after 7 players each made 10+ trips around the board Indiana Avenue had not yet been landed on. One of my friends insisted that lots of people were going to start landing on it because of the law of averages. At the time it seemed like good logic, but now I'm not so sure because that sounds a lot like the Gambler's Fallacy.

I'm not a master of this stuff so I'm probably missing something easy, but can someone enlighten me?

Thanks.

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At what point is it safe to say that an event is more likely to occur because it has not occurred as much as expected over a long enough period of time, even though mathematically the probability of an individual trial remains the same?


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Never. The difference is saying that something will happen versus saying when it will happpen.

[manpower]

Your friend was either joking or ignorant. The contradiction would only exist if infinity was a number that could be reached.

[ThinkQuick]

This may help... I'm definitely stealing it but can't recall the source.. so if anyone does please post it.

Lets say you sit down at a fair $1 coinflipping game and proceed to lose $10.
It would be a fallacy to say that since the game is fair, you will end up very close to even in the long run (By the law of large numbers (law of averages))

You are not permitted to determine when to start counting from, but must start on the next flip.

This means that now, when you are at -$10, the law of averages says that you will be very close to -$10 if you continue playing for a long time.

The point of the demonstration is to illustrate once again that past events do not affect future outcomes in these situations, and you must look at the problem anew from where you are now.

[SumZero]

First things first:

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I was playing a game of Monopoly the other day, and after 7 players each made 10+ trips around the board Indiana Avenue had not yet been landed on. One of my friends insisted that lots of people were going to start landing on it...

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I don't know about lots, but in the early game (when people try to get out of jail fast) it is the 13th most ended up on square with people ending their turn there 2.7357% of the time. In the late game (when people try to stay in jail as long as possible) it falls to 14th with 2.5671% of turns ending there. It is a pretty decent monopoly set to own (although not as good as the oranges!).

If we assume that each player takes an average of 4 turns to make a trip around the board than 7 players making 10+ trips is about 280 turns. And the probability of no player ending their turn on Indiana is about 1 in 2373. And if you consider doubles that don't end a persons turn there it makes it even more unlikely. So what you saw was quite unusual, but by no means impossible.

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... because of the law of averages.

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This part is obviously wrong.

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At the time it seemed like good logic, but now I'm not so sure because that sounds a lot like the Gambler's Fallacy.

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Your doubt is well founded as this is in fact the Gambler's Fallacy.

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How do the two ideas NOT condtradict each other? At what point is it safe to say that an event is more likely to occur because it has not occurred as much as expected over a long enough period of time, even though mathematically the probability of an individual trial remains the same?

I'm not a master of this stuff so I'm probably missing something easy, but can someone enlighten me?

Thanks.

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To understand the problem in the reasoning switch from monopoly to simple coin games.
Assume we know for a fact the true probability of an event occuring (for instance a fair coin which is 50/50 heads/tails).

Intuitively which of the following is more expected and which is less expected (given we know for a fact we have a fair coin):

Getting 7 heads and 3 tails in 10 flips.
Getting 70 heads and 30 tails in 100 flips.
Getting 700 heads and 300 tails in 1000 flips.

Hopefully you intuitively feel that the first is most common and the last is most surprising (and very good reason to suspect the coin wasn't fair). If we know the coin is fair the 7/10 case happens about 1 in 9 times. The 70/100 case happens about 1 in 43,000 (even rarer than your no Indiana situation). And the 700/1000 case happens about 1 in 2*10^37 AKA pretty close to never.

This is due to the law of large numbers, which states that a variable defined to be the sum of a series of independent and identically distributed random variables with a given mean will itself approach the mean times the number of terms in the series as the number of terms in the series gets large.

In other words if heads represents a 1 and tails a 0, then each flip is either a 1 or a 0. And if we define a variable to be a coin flip the mean value should be 1/2 or 50% for a fair flip. Now when we combine flips we can define a new variable (# heads/# flips), call it pHeads, and this variable has an expected value of 1/2 or 50% just like our single flip variable.

Now all three of my senarios involved represent a pHeads value of 70%, but with a different number of flips, and hence some were more likely to happen (the ones with fewer flips).

Ok, so now how does this relate to the Gambler's fallacy?

Imagine that you and I were going to flip fair coins in 10 flip chunks. But 10 different people were going to observe a 10 flip chunk each without knowing anything about the other outcomes. If you asked each person how many heads they expect to see, they should say 5. If you ask them how many heads they expect to see in the whole 10 sets of 10 they should say 50. That's the expected value and also the most likely value.

But now say we have knowledge of the first set of flips and we see that 7 out of 10 of the flips were heads (but we are still convinced we have a fair coin). Now the next person who comes in expects to see 5 heads on their set of 10 flips and expects to see 50 heads overall as most likely because they are unaware of the first set and are starting from a baseline of 0. But we have the knowledge of what happened in the past. And we now do NOT expect that 50 heads is the most likely outcome. Because the most likely outcome is 7 heads (that we've observed) plus the most likely outcome over the next 9 sets or 90 coin flips. That is 7 heads plus 45 heads. So we, because we know what just happened and aren't starting at a baseline of 0, know that at that moment in time the most likely number of heads to end up with is 52.

Note that 7/10 from our first set represents a pHeads value of 70%, but our new current prediction for most likely pHead value after 100 total flips is 52% once we've observed the 7/10 on the first set. And notice that 52% is much closer to 50% than our current observed 70%. That is, the reason that the percent of heads that gets flipped moves towards the overall average as we increase our trials isn't because tails are now "due" or are now more likely to come up, but because over our first 10 flips we were getting 70% heads and over all future flips we expect to get 50% heads. And (10 * 70% + X * 50%)/(10+X) approaches 50% as X gets larger.

Now imagine we see a second set of coin flips, this time with 6/10 heads. Now as our 3rd observer comes in with no knowledge of the past flips they start at the baseline of 0 and if they were to make a prediction on what pHead value they'd expect to see over the 100 flips their best guess would be 50%. But we've seen 13/20 heads which is a pHeads value of 65% that we've observed to date. Our new best current prediction for the final pHeads value after 100 flips given the 20 observations we have to date is now 53% (13 + 80/2). Again that is observed pHead value to date 65%, predicted pHead value after 100 flips including our observered 20 53%, actual "true" expected pHead value at the beginning or if we were to flip infinity times is 50%.

Now if on the next set we see 3/10 heads we'll get 16/30 heads for an observed pHeads value of 55% to date, and our new predicted pHeads value for 100 total including the observed 30 would be 51%. Etc.

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But now say we have knowledge of the first set of flips and we see that 7 out of 10 of the flips were heads (but we are still convinced we have a fair coin). Now the next person who comes in expects to see 5 heads on their set of 10 flips and expects to see 50 heads overall as most likely because they are unaware of the first set and are starting from a baseline of 0. But we have the knowledge of what happened in the past. And we now do NOT expect that 50 heads is the most likely outcome. Because the most likely outcome is 7 heads (that we've observed) plus the most likely outcome over the next 9 sets or 90 coin flips. That is 7 heads plus 45 heads. So we, because we know what just happened and aren't starting at a baseline of 0, know that at that moment in time the most likely number of heads to end up with is 52.


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"And we now do NOT expect that 50 heads is the most likely outcome." This statement is wrong: before the sequence you observed there could have been a series of 500 (or whatever) coins flips, all tails.

If this is not an issue, then you accept implicitely that one sequence of events (the first one) can be biased, but not any other (inlcuding a longer one)???

Your arguments only holds with big numbers and they are numbers that get close to infinity, not the numbers you are using.

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But now say we have knowledge of the first set of flips and we see that 7 out of 10 of the flips were heads (but we are still convinced we have a fair coin). Now the next person who comes in expects to see 5 heads on their set of 10 flips and expects to see 50 heads overall as most likely because they are unaware of the first set and are starting from a baseline of 0. But we have the knowledge of what happened in the past. And we now do NOT expect that 50 heads is the most likely outcome. Because the most likely outcome is 7 heads (that we've observed) plus the most likely outcome over the next 9 sets or 90 coin flips. That is 7 heads plus 45 heads. So we, because we know what just happened and aren't starting at a baseline of 0, know that at that moment in time the most likely number of heads to end up with is 52.


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"And we now do NOT expect that 50 heads is the most likely outcome." This statement is wrong: before the sequence you observed there could have been a series of 500 (or whatever) coins flips, all tails.

If this is not an issue, then you accept implicitely that one sequence of events (the first one) can be biased, but not any other (inlcuding a longer one)???

Your arguments only holds with big numbers and they are numbers that get close to infinity, not the numbers you are using.

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It seems like you completely missed the entire logic of SumZero's post. If you know a coin will be flipped 100 times, you expect to get 50 heads. If the first ten flips result in 7 heads, you now expect to get 45 heads on the remaining 90 flips, plus the seven you already have, for a total of 52 heads.

What happens before the observed sequence does not matter at all. All that matters is that we have a fair coin and a set number of trials. We can then adjust our estimation of the final number of heads based on the total number of heads so far, which would look like: (number of heads) + ((50%) * (remaining number of flips)).
As you should be able to see, this fits exactly with what is in SumZero's post.

You are correct in what you say. Perhaps I did misunderstand what was posited.

[AaronBrown]

The ideas do not contradict because averages regress to the mean, totals wander without preferred direction. The Gambler's Fallacy is to apply regression to totals of independent events.

For example, after 280 rolls in Monopoly, the average square has been landed on 7 times (I'm ignoring some fine points about squares that direct you to other squares like Chance and Go To Jail, and rolls when you are in jail). Indiana has been landed on 0 times, which is also 0% of the rolls.

If you continue playing, the percentage of rolls resulting in someone landing on Indiana will tend to go up until it reaches its long-term average of 2.5% (again, ignoring the fine points that lead to unequal square probabilities). At that point, it will go up or down randomly, with equal probability. However, the total number of times someone has landed on Indiana, which is now 7 below its mean, has no tendency to go up or down (ignoring the fine point that you're more likely to land on Indiana if you know you aren't starting from Indiana). The expected deficit of Indiana landings after 1,000 or 100,000,000 rolls is 7.

This assumes that all moves are independent. There are things that regress toward the mean even in total. If interest rates are high relative to recent history they are more likely to go down than up. If you parent is taller than average, you are likely to be shorter than your parent (but taller than average). The baseball player with the highest batting average at the All-Star break is likely to have a lower average in the second half of the season than the first. These are all regressions different from the tendency of averages to move toward the long-term mean.

[ThinkQuick]

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For example, after 280 rolls in Monopoly, the average square has been landed on 7 times (I'm ignoring some fine points about squares that direct you to other squares like Chance and Go To Jail, and rolls when you are in jail). Indiana has been landed on 0 times, which is also 0% of the rolls.


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Your response is helpful, but I do want to point out that due to dice rolling probabilities, jail, and the movement cards, the probability of landing on a space can vary quite dramatically between spaces, as you can see in this graph.