I'm new to statistics and I'm struggling with a hypergeometric problem:

Suppose I have a drawer with 100 socks in different colours, and suppose I know exactly how many of each colour there are. Now suppose I draw a handful of socks, say 10, at random. How do I calculate the probability of drawing a specific combination of colours?

If I draw, say, 10 socks, the site calculates what the odds are that, say, 5 of those are yellow. But how do I calculate the odds of drawing, say, at least 1 yellow, AND at least 2 blue, AND at least 3 red?

And what if I add further clauses such as: If there is only one yellow sock, it must not be drawn as the tenth?

How do I attack this problem?

Using the site, I can get the probability that at least 1 of the 10 socks is yellow OR that at least 2 of the 10 are blue OR that at least 3 of the 10 are red. I need all three to happen simultaneously, so I multiply the three probabilities...
- but this seems wrong, as the probability that at least 1 is yellow includes the case where, say, all 10 are yellow, and that conflicts with the other 2 requirements.
So I try to make room by having the site calculate the probability that 1-5 are yellow, multiplying that with the probability of 2 blue and 3 red...
- but 2-6 blue and 1 yellow and 3 red is also fine, and if I calculate that I get a different number...
- and 1 yellow, 2 blue, 3 red, and 4, say, green is also fine, and that produces a third probability...
- and I can't multiply them, as they aren't independent.

I've also tried to read the tutorial on the site, but I'm still confused. How do I solve this problem?

In short:

I draw ten socks. I need at least 1 yellow (which must be among the first 9 drawn), at least 2 blue, and at least 3 red, and I don't care about the rest. How do I calculate the odds of success, if I know there are, say, 10 yellow, 20 blue, and 30 red in the drawer (and 40 of other colours)?

And please, I need a method or a tool, so I can solve this kind of problem no matter what number of yellow, blue or whatever socks I need.

I'm counting on your expertise!

MindBoggle

Last edited by MindBoggle; 07-10-2016 at 02:23 AM.
Reason: ambiguity

I've managed to find two formulas. One goes C(n,r)=n!/r!(n-r)!, and allows me to calculate the total number of possible combinations of socks, drawing 10 from a drawer of 100. The result should be 100!/10!(100-10)! = 17.310.309.456.440.

So far so good. Now, how do I figure out how many of those combinations satisfy the requirements?

Maybe 10Choose1 x 20Choose2 x 30Choose3? Using the same formula, I get 10 x 190 x 4060 = 7.714.000 combinations = 0,000004% chance? No - that's gotta be wrong.

The other formula I found is more complicated, and goes: p=(kCx)((N-k)C(n-x))/(NCn) where k is the number of "successes" in the population, x is the number of "successes" in the sample, N is the size of the population, n is the number sampled, p is the probability of obtaining exactly x successes, and kCx is the number of combinations of k things taken x at a time.

This is the hypergeometric formula used on the site I linked to in the original post; I'm just not quite sure how it helps me. I think what I need to do is to calculate the number of possible 6-sock samples that contain exactly 1 yellow, 2 blue and 3 red, as after that everything becomes irrelevant (because I already have what I need a no further draw can change that). BUT - the extra, irrelevant draws can be anywhere among the first 10 (if, for a moment, I ignore the last requirement that there must be at least one yellow among the first 9 drawn). Hmmm...

Alright - going to bed. I'm gonna try to progress with this tomorrow.