[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 613
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Help me solve a problem:
There are 10,000 balls in a box. 50% of them are black and 50% are white.
We take 120 balls out of the box at random.
What is the probability that at least 30% of the balls taken out are white?
This task refers to trading! In general... one might think.
Do the balls go back in the box or not?
Yeah, I don't know what I'm talking about. Since when can trades be returned to the broker...
P.S. On a rough guess, that's about it. Balls taken out almost do not affect the ratio of probabilities 50 to 50 (they are few, and they are taken out about the same ratio). We get a classical Bernoulli scheme of 120 symmetric trials with p=1-p = 1/2, which must have at least 30 successes. There's a partial binomial sum there :(, I don't know how to calculate it quickly. Only an estimate.
But the probability is definitely very close to 1, since the probability that there will be less than 30 successes out of 120 at p=1/2 is almost vanishingly small. The S.Q.O. is sqrt(npq) = sqrt(120*1/2*1/2) ~ 5.5, so a deviation of 5.5 sigmas is an extremely rare thing.
No trading. Pure theorizing :)
No balls in the box.
Yes, let's assume that the ratio is always 50/50, it's probably easier that way. Or let it be 100000 balls in the box, doesn't matter.
I've already answered that. Practically one - with a variation of no more than a thousandth of a percent.
For example, if I need not 120, but a smaller number, not 30%, but a larger number.
For example, a function of this kind:
Probability = Function (How many balls were taken out, Minimum fraction of balls);
If the exact formula is
p=Sum( C(120, k) * p^k * (1-p)^(120-k); k = 30..120 )
If approximated, there is a limit theorem: with a large number of trials n (here 120, already quite large; the criterion for "large" n is np(1-p) > 5) the binomial distribution tends to the Gaussian N(np, npq). Accordingly, it remains to calculate in any statistical package (or even in Excel) the Gaussian integral. The limits of integration are roughly from (120*p-30)/sigma to + infinity (here).
Sigma = sqrt(npq).
p=Sum( C(120, k) * p^k * (1-p)^(120-k); k = 30..120 )
Sum - sum, C - combination
Well p to the left of the equal sign is different, of course. Well, let P.
C(n,k) is the number of combinations of n by k, i.e. in common parlance, the binomial coefficient.
Sum is simply the summation, in this case by k.
Well, in short, it's a long explanation, if you don't know. This is a terver, and by no means its most complex sections.
Dima, why do you want to know the probability that differs from one in thousandths of a percent? If you want guarantees, there are none. Nobel laureates (LTCM) and Niederhoffer himself covered themselves with probabilities to some degree minus one - and still "hit".