In a dark room there is a box with 8 red and 5 black socks

Question

In a dark room there is a box with 8 red and 5 black socks

1. what is the minimum number of socks you need to take to guarantee a pair of the same color ?
2. what is the minimum number of socks you need to take to guarantee a pair of red socks ?
3. and if you need to make a pair of red and a pair of black socks ?
4. 7 socks are removed from the box described in the previous problem what is probability that this sample will have exactly one pair of black socks ?

Answer 1

I found answer
If you pick three socks, you will have either a red or black pair, that is the minimum needed to guarantee a pair. Pick one sock, you don’t have a pair. Pick a second sock, it could be the other color from the first one. Pick a third, worst case scenario you have one of each color, plus one more that matches one of the two you already have.

However, you need to pick seven socks to guarantee that you have a red pair. If you pick five or less, they could all be black, if you pick six you could have five black and one red, only by taking the seventh sock can you guarantee, even if you’ve picked all five black, that you MUST have two red.
this answer by Lance Berg

Answer 2

HINT To "what is the minimum number of socks you need to take to guarantee a pair of the same color ?"

Think about how you get the number of possibilities to get two red socks and to get two black socks and how much different possibilites do you have at all get any combination.