# Probability of socks selection

I don't even know how to start this question:

I have 6 pairs of socks (12 socks) of different colors.

If I choose 4 socks randomly, what is the probability that I will pick 2 socks of same color, and the other 2 are different colors?

chiro
Hey there webing86 and welcome to the forums.

Have you made an attempt at answering the question? If so could you show whatever you've tried so that we can help you see what you understand, and that you have attempted the problem on your own (its the forum policy to do this).

In saying that, what have you identified as your sample space and the appropriate subset for any or all of the events? (ie probability of one sock any color, two socks same color, two socks same with one with other color, or all four socks)?

Its best to break up your probability space into its simplest events and then use probability axioms to figure out the total space of the event.

Hey there webing86 and welcome to the forums.

Have you made an attempt at answering the question? If so could you show whatever you've tried so that we can help you see what you understand, and that you have attempted the problem on your own (its the forum policy to do this).

In saying that, what have you identified as your sample space and the appropriate subset for any or all of the events? (ie probability of one sock any color, two socks same color, two socks same with one with other color, or all four socks)?

Its best to break up your probability space into its simplest events and then use probability axioms to figure out the total space of the event.

Hi,

I have tried to solve this by narrowing my sample size.

I used 3 pairs of socks (red, red), (blue, blue), and (yellow, yellow).

The combination of picking 2 socks of same or different colors will be 6 (rr, rb, ry, bb, by, yy). I can list out all the combinations as the sample size is small, but I just do not how to derive it using formula.
* If all the socks are of different colors, of course the combination will be 6C2.

chiro
Hi,

I have tried to solve this by narrowing my sample size.

I used 3 pairs of socks (red, red), (blue, blue), and (yellow, yellow).

The combination of picking 2 socks of same or different colors will be 6 (rr, rb, ry, bb, by, yy). I can list out all the combinations as the sample size is small, but I just do not how to derive it using formula.
* If all the socks are of different colors, of course the combination will be 6C2.

Its not a good idea to try and think of things in terms of formulas. Formulas can be useful if you have a system with some pretty common assumptions (i.e. there are a lot of real world phenomena that use these assumptions, so its handy to use that particular model), but its a lot better to use the foundational concepts and axioms of probability.

Having said that, there is a model known as the hyper-geometric distribution:

http://en.wikipedia.org/wiki/Hypergeometric_distribution

Your kind of problem looks like it can be solved using this kind of distribution. This distribution has a probability density function that is used to calculate probability given a total multivariate sample size (N categories of things - in your case sock colors) and from this you use the formula to get your probability.

If I were you I would try to find a derivation of the distribution and look at any assumptions that have been made to get to that result. Its probably a good idea to learn the binomial first and work your way up to the multivariate hyper-geometric distribution.

say the 6 colors are a,b,c,d,e and f.

then, the given cases will be:

aa-bc
aa-bd
aa-be
aa-bf
aa-cd
aa-ce
... and so on.

(1) 2 same color and 2 different colors : 6 * 5C2 = 6 * 10 = 60
(2) all the cases : 12C4 = 55x9

P = (1)/(2) = 60 / (55*9) = 4 / 33

I am not 100% sure though... :p