FleaCircus

My question is a little involved, but here goes:

Let's say you're flipping a coin--you have two possible outcomes, heads and tails. The likelihood of heads coming up once (in one toss) is 1/2, twice (in two tosses) is 1/4, three times (in three tosses) is 1/8.

So we can generalize that the chance of heads coming up n times in n tosses is 1/2^n.

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?

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. At what point does this potential for an outcome by chance alone become statistically negligible?

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.

And then I'd wear a tinfoil hat so that he couldn't read my mind.

Edit: I'm not trying to prove/disprove psychic abilities, I'm just using it as an example.

__________________
"We must have waffles. We must all have waffles, forthwith. Oh, we must think.
We must all have waffles and think, each and every one of us to the very best of his ability."
-- Professor Goldthwait Higginson Dorr, Ph.D.

Yakk

That's n times in a row, and correct.

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%.


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.

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.

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... ;-)

__________________
Last edited by JHVH : 10-29-4004 BC at 09:00 PM. Reason: Time for a rest.

FleaCircus

Awesome, thanks for the help.

__________________
"We must have waffles. We must all have waffles, forthwith. Oh, we must think.
We must all have waffles and think, each and every one of us to the very best of his ability."
-- Professor Goldthwait Higginson Dorr, Ph.D.

FleaCircus

I did a little more research, and this is what I came up with.

Taken from HyperStat Online

__________________
"We must have waffles. We must all have waffles, forthwith. Oh, we must think.
We must all have waffles and think, each and every one of us to the very best of his ability."
-- Professor Goldthwait Higginson Dorr, Ph.D.

zfleebin

Fleacircus: If you think the guy is psychic you should take him to vegas and forget the coins. No, seriously in statistics we use alpha and the null hypothesis to determine significance. The null or null hypothesis is the hypothesis of no difference. In the case of your friend the null hypothesis states that his ability to predict the side the coin will land on is no better than anyone else in the population. This is where alpha comes in. Alpha is usually set at .05 which means that he would have to correctly predict the side it landed on more than 95 times out of a hundred to reject the null hypothesis. If he did correctly predict the side the coin would land on 96 times out of 100 you would reject the null hypothesis and he could say that the results of his test throws supported his hypothesis that he was psychic. Just as an add on I have never seen any research that showed a psychic was significantly better at telepathy than the rest of the world.

rsl12

fleacircus: check out any basic college statistic textbook and look up the two-tailed student t-test. that will give you what you need to know about determining with confidence that your friend has something up his/her sleeve.

__________________
oh baby oh baby, i like gravy.