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I am interested in probabilities regarding roulette. I know only the most basic rules apply to this game and each number and spin of the wheel is independant. So I dont want to go into much detail cause my math skills are good, but my knowledge of the subject is limited.

I want to compare it to a coin toss scenario cause its easier for my little brain to understand. Assuming a fair coin. If tossed once has the same chance of landing heads(H) or tails(T) 50/50. Of course like roulette, each coin toss is indepandant of the next. I know this so there is no need to point it out.

However, stastically speaking, if we flip a coin an unlimited number of times, and somewhere in that number it decided to show 8 heads in a row (still assuming its fair), is there not now a much higher chance that the next one will be heads.

If not why, or if so why, either way how can I calculate the odds of this situation.

Before answering, consider one further point. If you as a mathematician, were forced to bet (life of death, literally no choice in the matter), given the conditions above, which side of the coin would you bet on, and why. Explain with maths if possible.

This is my edit: First thank you kindly for your answers. I have 3 so far, all of them doubt the fairness of the coin. Lets forget its a coin at all, I would even say put in a machine that is truly random, but then someone is going to tell me that it uses a formula and therefore cant be truly random, it is therefore psudo random.

So lets forget the coin and machine, and use a magical harry potter device with 2 sides that definitely always is truly random, no matter what.

Assume it is truly random and fair

Given an unlimited number of flips, this possibility will happen eventually. Imagine it did, and your life depended on the correct answer.

How would you bet and why.

One last edit (Independance):

Many people answering questioned the fairness of the coin. This as mentioned above is unquestionably that the coin must be considered fair.

One answer below by "tskuzzy", highlighted the key phrase "Independance". I never said anything about independance. In fact, I did make a point of saying an unlimited number of times and somewhere it decided to show 8 heads in a row.

Given the only thing in question is the independance of the throws, it is quite safe to say something like "well the last 8 times it came up heads, lets group it into a set of 10 throws which means it came up 8 out of 10 times heads, therefore twice only tails" Or even equally safe to group it such "The last 8 times it came up heads, lets group it into a set of 20 throws which means it came up 8 out of 20 times heads (assuming you know for sure this was the case. i.e. that no other heads were thrown prior to the 8 in a row)".

This brings probabability back into it. Which was the point of the question. Otherwise gambelers falacy theorm definitely applies. So does it still apply or not?

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You have been betting tails, and $8$ times in a row, the coin has landed heads. You have lost your money, then your watch, and, finally, your house. So on the $9$th toss, you bet your right arm. Do you believe that the coin thinks, poor fransasm, I better make it up to him/her, I will tilt towards tails? Probably the coin in fact doesn't care about you, and doesn't even remember what it came up with in the recent past. –  André Nicolas Aug 4 '11 at 0:02
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4 Answers 4

up vote 3 down vote accepted

If you believe the coin is fair and tosses are independent, the probability the next toss is heads is 1/2. As you say, in a long run you expect 8 heads in a row to happen occasionally, so it shouldn't change your estimate of the next toss. If the first 8 throws of a test come up heads, it might shake your faith that the coin is fair, so you have to consider where that belief comes from.

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Thanks for answering. But lets insist that the coin is definitely fair. In fact lets even remove the coin and put in a magical harry potter machine and pretend the are truly random (cause normal machines are only psudo random). How would you work out your odds of the next one being say heads. Is it still 50/50 or is it 8 of one side have come up in a row, shouldnt have happened, but did, so now its 9 (for the next flip) /8 (for the previous 8 flips) or something like that for being the same side and subtract from 1 for being the opposing side. Or is there some other mad way of doing it. :-) –  Francis Rodgers Aug 4 '11 at 0:11
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If the coin is fair, the odds are 1/2 and history doesn't matter. As tskuzzy says, that is the definition of fair. You might look at en.wikipedia.org/wiki/Gambler%27s_fallacy –  Ross Millikan Aug 4 '11 at 0:24
    
I read that before. And the infinate monkey theorm. That is the reason for the question. But you have given me the best answer so far. So I am going to vote for you. Cheers. –  Francis Rodgers Aug 4 '11 at 0:38
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This is a matter of information. If you know a priori that each toss is independent of the previous tosses, then of course the outcome of the next toss will be 50-50 even if the last 1000000 were heads. That's the definition of independence.

However if you did not know that the coin was fair and independent, then the problem becomes more interesting. Then it's a matter of statistics and inference. In this case, I would calculate the probability of heads and tails in my sample (the previous coin flips). Then I would pick the side with higher probability as the next outcome.

The reasoning behind this is simple. Suppose I flipped the coin 100 times and it came up heads 89 times. Then I have more reason to believe that the coin is weighted towards heads than tails. Now I say more reason because I can't be 100% sure. However there comes a point where I can be sure beyond reasonable doubt that this is the case.

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So would you bet next on heads or tails. If heads then why. if tails then why. Given your logic I would bet tails because the chances of the next one being heads again is growing ever smaller each time heads is flipped. Conversly we could play devils advocate here and say because it came up so many times one side, then the chances are high that this streak with continue. Either way, which would you choose if forced and can you explain why with some basic (or complicated) maths. Thanks for answering. –  Francis Rodgers Aug 4 '11 at 0:16
    
Also I would like to add to this. Now that you highligheted it. bold I never said the coin was independant, only that it was fair /bold So probabability does have a role to play here. Otherwise the question is pointless. Its the probabability I want to find out. In such a circumstance one might decide to group the last 10 flips and say if 8 were heads then the two before were tails, so there is x chance of one vs other. Once you start grouping things, you remove indepandance, so does that make a difference. This is the real question? Thanks for your answer. –  Francis Rodgers Aug 9 '11 at 11:46
    
"each coin toss is indepandant of the next" –  tskuzzy Aug 9 '11 at 11:47
    
2 to the millionth power is about 10 with 100,000 zeros. It would take far fewer than that many heads to convince anyone the coin isn't fair. I'd guess a few dozen and most people cry "foul" –  JoeTaxpayer Dec 13 '13 at 2:47
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This is known as the gambler's fallacy.

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The alternate argument to it is the infinate monkey theorm. My question is given knowledge of both. And the fact that the coin is definitely fair. How would such a run affect your bet "if at all", if you life depended on it, and why? Thanks for your answer. –  Francis Rodgers Aug 9 '11 at 11:38
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There are two incorrect assumptions that some people make. One is as you explained here, if you get $8$ heads in a row, then tails is overdue so it is more likely than $1/2$ that tails will occur next. As explained in another answer, this is incorrect. One also sees the opposite assersion: if there were $8$ heads in a row, then heads is in a streak or heads is hot so it is more likely than $1/2$ that heads will occur next. This is also incorrect.

Of course this answer assumes it really is a fair coin.

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Let's imagine a science fiction story. It goes like this. A sneaky guy working for a mobster goes to the bank and gets a large supply of newly-minted quarters. He takes them back to the hide-out. Flips them all. About half come up heads. He flips those again, about half of those (a 1/4 of the original number) come up heads a second time. And so on. After an hour he has this one quarter left that came up heads 10 times in a row. He knows it is a tails overdue quarter.

Now the mobster takes this out to the event he wants to bet on. Any scientist examining the quarter, analyzing it, etc., will find nothing different about it. But that mobster knows it is a tails overdue quarter. Imagine how he can get rich with such a special thing in his pocket! And he has a bunch of flunkees back at the hide-out busily creating more of them.

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I am beginning to be converted to this view, especially when given the undeniable fact that somehow no matter what, in this question, the coin must be fair. If not for my insistance on that fact we could argue all day. So you assert that because the coin is definitely fair, it would make no difference to your choice. Would be kina hard not to be swayed by emotion if your life depended on it though. But you definitely say it makes no difference to you. Thank you for taking the time to answer. –  Francis Rodgers Aug 9 '11 at 11:32
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