To elaborate a little on the previous answer. Number your urns 1 through 10, with the tenth urn being the one with 5 white and 1 black ball, and let P(10|W) denote the conditional probability that the ball came from the 10th urn, given that it is white.
From the definition of conditional probability, you know:
P(10|W) = P(W and 10th)/P(W)
You can compute the probability that it came from the tenth urn and is white, since this is just (1/10)(5/6). So the question is what is P(W)? But we can decompose this event as the sum of the disjoint events:
P(W) = P(W and 1) + P(W and 2) + .... P(W and 10)
Note that if n is not equal to 10 (i.e. n=1,2,...,9), then P(n and W) = (1/10)*(1/2), so
P(W) = (1/10)(9/2 + 5/6) = 32/60 = 8/15