Well, you've got two possible situations here:
a. you pick the blue marble first and then you pick the black marble. The probability of picking the blue marble first is 6/10 = 3/5 and the probability of picking the black marble second is 4/9 (4 desired and 10 - 1 = 9 possible). So the probability of the case is 3/5*4/9 = 4/15.
b. you pick the black one first and the blue one second. The probability of picking a black marble first is 4/10 = 2/5 and the probability of picking a blue marble is 6/9 = 2/3. So the probability of the case will be 2/5*2/3 = 4/15.
This means that the probability that one blue and one black marble will be picked without replacement is 4/15 + 4/15 = 8/15.