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[GendoIkari]

So lots of people here are really good with probability. Despite being a math major, it's not my strong suit. Anyway...

If you flip a coin 22 times, what are the odds that 19 of the flips will be heads? How about at least 19? My instinct says the answer is a really small chance of that happening; since you should expect only 11 heads.

[DStu]

Exactly 19 is 2^(-22)x(22 choose 19) =2^(-22)x22x21x20/3/2=2^(-22)x22x7x10 = 0.00036716461181640625

At least 19 is not much more... Just sum over exactky 19,20,21,22.

[Kuildeous]

This may be of some help to you:

http://www.mathwords.com/b/binomial_probability_formula.htm

[WanderingWinder]

You can also google binomial calculators.

On the other hand, most coin flips aren't actually independent trials, nor do they have p = .5.....

[Guy Srinivasan]

Another way to do it: a coin's expected value is 0.5 and variance is 0.25 and you want 18.5+ on 22 coins or (18.5-22*0.5)/sqrt(10*0.25) which is slightly less than 7.5/1.5 = 5, so you're looking at something more rare than a 4.5 standard deviation away from the mean event. a.k.a. << 1%.

[Grujah]

At least 19 is not much more... Just sum over exactky 19,20,21,22.

Or easier, sum for exactly 0,1,2,3 heads and substitute that from 1

[Ozle]

Is the answer 6?

[Davio]

Is it an imaginary coin or a real coin?

Real coins don't exactly have a 50/50 distribution, more like 51/49, depending on the weight distribution and what face starts facing up when tossing it.

 

[Eevee]

You can also google binomial calculators.

On the other hand, most coin flips aren't actually independent trials, nor do they have p = .5.....

Interesting, care to elaborate?

[WanderingWinder]

Mostly, it is what Davio has said - the probability of what comes up is non-negligibly impacted by how it was before you flipped the thing. 51% has been found in one study, that got repeated a bunch of places. But other studies have found it to be bigger. It really depends on the person who is doing the test, how they flip. Some people will have the flip-to-flip correlation of upwards of .6. In general, you get some kind of correlation, anyway.

Unless you meant the first part, in which case: http://stattrek.com/online-calculator/binomial.aspx

[Grujah]

51/49 is what I've found in a Stanford study as well.

Or if you read American Gods, you can rig the flip with 100/0 chance.

[DStu]

Or if you read American Gods, you can rig the flip with 100/0 chance.

Or if you are Persi Diaconis: http://news.stanford.edu/pr/2004/diaconis-69.html (long but interesting anyway)

[DStu]

Unless you meant the first part, in which case: http://stattrek.com/online-calculator/binomial.aspx

Even if you mean the first parts, you can get some bounds for also p!=0.5 there.

With p=0.6, at least 19 turns would be 0.75%.

And 0.6 is pretty extreme I would say, unless you train to throw coins in biased ways. Especially if you don't purpusely start with the same side up.


Btw, is there a problem with this shady tactics: I train to throw coins in biased ways. The rulebook never stated in which way I have to throw the coin, so I told all my opponents they should throw the coins in such a way that they most of the times show heads...

[Kirian]

Or if you read American Gods, you can rig the flip with 100/0 chance.

Or if you are Perci Diaconis: http://news.stanford.edu/pr/2004/diaconis-69.html (long but interesting anyway)

Thanks for that!  I'd read about Diaconis's work before (and I'm pretty sure he was discussed in one of the many shuffling threads) but I'd never read that story.  Sounds like a cool dude.

Btw, is there a problem with this shady tactics: I train to throw coins in biased ways. The rulebook never stated in which way I have to throw the coin, so I told all my opponents they should throw the coins in such a way that they most of the times show heads...

Easier just to carry around a coin which has heads on both sides.