[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.