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Originally posted by FleaCircus
So we can generalize that the chance of heads coming up n times in n tosses is 1/2^n.
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That's n times in a row, and correct.
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But how can we generalize the chance of heads coming up any x times in n tosses, where x<n? For example, what's the probability of any 7 of 10 tosses coming up heads?
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It works out to
10C7 * 1/2^10
IIRC.
10C7 is the number of ways you can choose 7 elements out of a list of 10 elements.
It works out to
10!
---------------
(10-7)! * 7!
where n! = 1*2*3*...*n
In this case, it is
10*9*8
----------
3*2*1
which is 120.
1/2^10 * 120 is about 12%.
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Next, let's say that I have a friend who claims he's psychic, and can predict the outcome of a coin toss. If he gets 7 out of 10 tosses right, that's pretty good, but the generalization above gives us the probability of such an outcome happenning by random chance alone.
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In reality, you care about the chance he got 7 8 9 or 10 right (you would be no less impressed by 8 or 9 or 10). That's
(10C7 + 10C8 + 10C9 + 10C10) * 1/2^10
=
(120 + 45 + 10 + 1) * 1/2^10
=
176 * 1/2^10
= approximatally
17%
So, getting 7 coins right is about a 1 in 5 occurrance.
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At what point does this potential for an outcome by chance alone become statistically negligible?
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Statistical negligibility isn't a hard and fast point. How important is it?
However, what you care about is:
What is the chance that his ability to predict coins more accurate than 50%?
My statistics is too rusty to answer that one.
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I know that, even with a million tosses, there's still an outside chance that he could get them all correct by randomly guessing. But it's so slim a chance that, if he could get that many right (and especially if he could repeat it), then I'd presume that chance alone did not dictate the outcome.
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Repeating it is quite important.
If there is a 1/10 chance he can do it once, there is a 1/100 chance he can do it twice, and a 1/1000 chance he can do it 3 times.
Of course, assuming he can cheat somehow is still a good hypothesis. Cheating at coin flips is surprisingly easy, unless you are careful.
A friend of mine used to do a neat card trick.
He'd get someone to shuffle a deck of cards, then select one card, look at it, and put it back.
Without touching the deck, he'd then try to guess what the card is.
By guess, I mean, he'd pick a random card, and say "was it a 7 of clubs?"
1 / 52 he was right, and the person he pulled the trick off on was very impressed... ;-)