Re: For the mathematically minded
 

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Wow what a wonderful contribution.

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I assume that's sarcastic, although it's hard to tell in message postings. I apologize if I'm wrong.

It is an important contribution. Of all the coins in the history of the world that have been flipped 21 times heads in a row, I'll bet none of them were fair. When you see a highly unusual event, it's very likely that the assumptions underlying the computation were wrong. If a coin flips 21 heads in a row, bet on heads next flip.

As a theoretical calculation, 0.5^21 is correct. But it's important to state the assumption.

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When people talk about probability problems involving coins they always mean fair coins unless they specify otherwise. It's not close.

Re: For the mathematically minded
 

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When people talk about probability problems involving coins they always mean fair coins unless they specify otherwise. It's not close.

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Except when WPT announcers talk about the QQ vs AK "coin flip." Talk about a ridiculously unfair coin.

-RMJ

Re: For the mathematically minded
 

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Assuming "fair" means "between .495 and .505" or some other suitable approximation, I'll take that bet. 5 in 10,000,000 isn't so unlikely that it's never happened in the history of the world.

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I won't give you the 0.495 to 0.505, what I mean is I would bet that the coin has greater than 0.5 chance of coming up heads.

It's true the world has been around for a while, but how often does someone flip a coin 21 times in a row? It could well be in the millions, or billions, in which case you're probably right that someone hit 21 heads in a row with a coin that had 0.499999 chance of coming up heads. I got a bit carried away. Let me reduce my bet to, if I saw a coin come up heads 21 times in a row, I would bet that it had greater than 0.5 chance of coming up heads.

Re: For the mathematically minded
 

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When people talk about probability problems involving coins they always mean fair coins unless they specify otherwise.

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I disagree on two levels. Some people (as RocketManJames points out) use "coin flip" to mean any binary outcome over which no one has any control or way of getting information other than doing it. Going all in heads-up preflop is a coin flip, the way many people use the term, but that doesn't mean the odds are even.

Second, even when people always mean something, it can be useful to point out the underlying assumptions.

Re: For the mathematically minded
 

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Second, even when people always mean something, it can be useful to point out the underlying assumptions.

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See, this is why I love you (not that way). I spend all day getting ridiculed by my friends for being a boring, anal actuary. You're not even an actuary, and you're just as bad.

Re: For the mathematically minded
 

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Of all the coins in the history of the world that have been flipped 21 times heads in a row, I'll bet none of them were fair.

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I think "none" is a bit strong, although its not a bet we would be empirically able to resolve.
I take the general point about there being an increased likelyhood of bias either in the coin or its flipping.
The likelyhood of any such bias however is also determined by many other factors.

Analogous example:

Home game: my opponent is dealt two straight flushes and a royal flush. I am dealt the ace high flush twice and a full house the third time. This happens in three consequetive occaisions when my oppenent is dealer.

Do I suspect he might be cheating?

Party poker 1/2: same thing happems. Evem down to the dealer button being with him on each occaision.

Do I think Pary is rigged, or that he has cracked their software?

Re: For the mathematically minded
 

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Let me reduce my bet to, if I saw a coin come up heads 21 times in a row, I would bet that it had greater than 0.5 chance of coming up heads.


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Really? I think I would bet it was a two-headed coin, and the chance of it coming up heads again was slightly greater than .5, like say, 100%
[img]/images/graemlins/grin.gif[/img]

Re: For the mathematically minded
 

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Of all the coins in the history of the world that have been flipped 21 times heads in a row, I'll bet none of them were fair.

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I think "none" is a bit strong, although its not a bet we would be empirically able to resolve.
I take the general point about there being an increased likelyhood of bias either in the coin or its flipping.
The likelyhood of any such bias however is also determined by many other factors.

Analogous example:

Home game: my opponent is dealt two straight flushes and a royal flush. I am dealt the ace high flush twice and a full house the third time. This happens in three consequetive occaisions when my oppenent is dealer.

Do I suspect he might be cheating?

Party poker 1/2: same thing happems. Evem down to the dealer button being with him on each occaision.

Do I think Pary is rigged, or that he has cracked their software?

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"none" is absolutley not too strong a term. It is physically impossible for a coin to be precisely fair. Also, atmospheric effects are constantly changing, so even if a coin could somehow be fair on one flip, it would not be fair on the next.

Re: For the mathematically minded
 

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"none" is absolutley not too strong a term. It is physically impossible for a coin to be precisely fair. Also, atmospheric effects are constantly changing, so even if a coin could somehow be fair on one flip, it would not be fair on the next.

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lol. Your response reminds me of the post asking weather the probability of God existing can change. (of course it can).

Probability and fairness are observer based models. I can decide that the guy who three bet me on the river has a 90% chance of holding QJ to make the nut straight. But that 90% only applies to my POV. from his POV he is 100% certain one way or another.

Fairness on a single flip depending on gusts of wind and random changes in throwing angle etc. are not relevant unless known to the observer in advance. The coin will of course flip one way or the other, having all the data to hand in principle (though not in practice) you could predict the outcome. The real question is wether when you control for (i.e randomise) all the other factors does the coin under a series of flips display any bias.
I suspect factors like throwing technique, throwing height, landing surface all have such a large impact that any inherent bias in the coin would be effectively masked from detection. Also the direction of any bias under specific conditions may well reverse under alternate conditions.

I am willing to accept that the design of a coin (weight distribution of the embossing perhaps) may potentially bias a coin under certain condtions.
But would that make the toss less than fair, given that this is a 'coin toss' and not a disk toss.

In any case to discern any gerneral bias would require a large number of trials say 1000 different people each tossing the coin with gloves on and blindfolded 100 times in different conditions. Even then youd only end up with a probability as to wether or not the observed bias was significant.

More likely IMHO is the observaton of bias with a particular tosser in a specific location with a particular type of coin, irrespective of the actual coin used.

Any additional concern I might express with regard to a large sequence of heads would thus be put down almost exclusively to those factors rather than to some inherent defect in the particular coin.

Re: For the mathematically minded