joe had 4 red sock and 2 green sock and everyday he pulled two sock
out of the draw with his eyes closed. how many times will he pull out
socks that do not match?

When you said, "how many times will he pull out socks that do not
match?", I assume you meant, what is the probability that two random
socks pulled out of the drawer will not match.  I also assume that the
drawer is refilled with the six socks each morning.  I also assume you
don't fold matching socks together in the drawer.

In that case the probability of pulling out two random socks that do
not match is 8/15.

There are two possible cases that need to be considered

1.  Pull out a red sock then a green P(G|R) = (4/6)*(2/5) = 8/30 = 4/15
2.  Pull out a green sock then a red P(R|G) = (2/6)*(3/5) = 8/30 = 4/15

Probability = P(G|R) + P(R|G) = 4/15 + 4/15 = 8/15.

A correction:

In that case the probability of pulling out two random socks that do
not match is 8/15.


2.  Pull out a green sock then a red P(R|G) = (2/6)*(4/5) = 8/30 = 4/15