For the mathematically minded

Hi all, just a quick probability question that I'm not sure about. What are the odds of flipping a coin and having it come up heads every time 21 times in a row?

[KJL]

.5^21= ~.00005%

[SheetWise]

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Hi all, just a quick probability question that I'm not sure about. What are the odds of flipping a fair coin and having it come up heads every time 21 times in a row?


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FYP

[jason_t]

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Hi all, just a quick probability question that I'm not sure about. What are the odds of flipping a fair coin and having it come up heads every time 21 times in a row?


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FYP

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Wow what a wonderful contribution.

The probability of any fixed particular sequence of length 21 of heads and tails is (1/2)^21 = 4.76837158 × 10-7.

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Wow what a wonderful contribution.

The probability of any fixed particular sequence of length 21 of heads and tails is (1/2)^21 = 4.76837158 × 10-7.

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Not if you're talking about normal coins.

P.S. .5^21 = (1/2)^21

[AaronBrown]

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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.

[Jerrod Ankenman]

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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.

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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.

Now, I'll give you a fun example of a claim that hasn't happened in the history of the world. Someone once claimed to me that they were watching for AQ vs AK showdowns on PokerStars all in preflop and made a spreadsheet where they marked the result. They claimed that after 2000 trials, AQ had won 50%. Bill and I estimated the probability of this happening. It was like 1 in all the electrons in the universe.

(obviously, either the site is rigged which i doubt or there's a little selection bias problem here)

jerrod

[SheetWise]

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Wow what a wonderful contribution.

The probability of any fixed particular sequence of length 21 of heads and tails is (1/2)^21 = 4.76837158 × 10-7.

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Jason_T

You surprise me. Usually you surprise me because I'm wrong. I have great respect for both your talents and insight -- but do you really see this correction of an assumption as trivial?

[fluorescenthippo]

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Hi all, just a quick probability question that I'm not sure about. What are the odds of flipping a fair coin and having it come up heads every time 21 times in a row?


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FYP

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it is very obvious he is talking about a fair coin.

sorry you wasted your FYP

[Siegmund]

So, now that this question has come up...

How many fans of Tom Stoppard's Rosencrantz and Guildenstern are Dead do we have in the house?

[jason_t]

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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.

[RocketManJames]

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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

[AaronBrown]

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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.

[AaronBrown]

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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.

[mosdef]

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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.

[malorum]

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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?

[cardcounter0]

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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%
/images/graemlins/grin.gif

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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.

[malorum]

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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.

[malorum]

http://www.maa.org/mathland/mathtrek_03_01_04.html

[malorum]

http://www.sciencenews.org/articles/20040228/fob2.asp

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Their preliminary data suggest that a coin will land the same way it started about 51 percent of the time. It would take about 10,000 tosses before a casual observer would become aware of such a small bias, Diaconis says. "Maybe that's why society hasn't noticed this before," he says.

This slight bias pales when compared with that of spinning a coin on its edge. A spinning penny will land as tails about

80 percent of the time, Diaconis says, because the extra material on the head side shifts the center of mass slightly.


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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.

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You're completely, absolutely right. My previous comment was not well thought-out; yours was.

Still, I believe the probability that any given coin is "fair" (ie, the coin has precisely a 50% chance of coming up heads on a given flip) has to be zero. This stems from the fact that probabilities are on a continuum from 0 to 1, and probability .5 is only a discrete point on that continuum.