Stinky Socks

Let Ram pick first, (it doesn't matter which color at this point). Now there is a probability of $\frac{1}{7}$ that Shyam will pick the other sock of the same color that Ram chose, in which case no matter what their second picks are neither will end up with a matched pair.

Now suppose Shyam picks a sock of a different color than his brother picked, an event that will occur with probability $\frac{6}{7}$ . Then if Ram picks the other sock of the color Shyam first chose, (an event that will occur with probability $\frac{1}{6}$ ), then neither will end up with a matched pair no matter what Shyam's second pick is. If, however, Ram's second pick differs in color from the first two chosen, (an event that will occur with probability $\frac{4}{6}$ given that Shyam's first pick was of a different color than Ram's first pick), Shyam will have a probability of $\frac{4}{5}$ of choosing a sock that does not match his first pick.

So putting these pieces together, the probability that neither brother ends up with a matching pair of socks is

$\dfrac{1}{7} + \dfrac{6}{7}*\dfrac{1}{6} + \dfrac{6}{7}*\dfrac{4}{6}*\dfrac{4}{5} = \dfrac{2}{7} + \dfrac{16}{35} = \dfrac{26}{35} = 0.743$ to $3$ decimal places.