Statement 1: The probability is less than 0.2 that the first sock is black.
Thus, (B/8) < 0.2 => B < 1.6
As B must be a non-negative integer, possible values of B are 0 and 1.
1. If B = 0: Required probability = 0
1. If B = 1: Required probability = 0

Sufficient.

Statement 2: The probability is more than 0.8 that the first sock is white.
Say number of white socks = W, then (W/8) > 0.6 => W > 6.4
As W must be a non-negative integer, possible values of W are 7 and 8. Thus the number of black socks is either 0 or 1. The rest of the analysis is same as statement 1.

Rahul@gurome wrote:
As I have mentioned in the "Given" analysis, the probability that both socks are black = (The probability that the first sock is black)*(The probability that the second sock is black) = (B/8)*((B - 1)/7)

This is because...

• Probability that the first sock is black = (B/8)
  Probability that the second sock is black = ((B - 1)/7)

Hope this is okay with you.
Now put the possible values of B in the expression. For B = 0 and B = 1, the probability is zero!

(This can be shown without introducing any formula. If the number of black socks is zero then the probability that both socks are black is obviously zero as there is no black socks at all! If the number of black socks is one then also the probability that both socks are black is obviously zero as after the first pick there is no black socks!)

Rahul@gurome wrote:Remember that total number of socks is 8. Now there are either 7 or 8 white socks.

If there are 8 white socks then all the socks are white => No black socks.
If there are 7 white socks then only one sock is non-white => Number of black socks = 1 (If the remaining sock is black) or Number of black socks = 0 (If the remaining sock is of some other color)

Thus number of black sock is either 0 or 1.