if a black ball is drawn for the 4th time on the 7th draw that means in the preceeding 6 draws, we have drawn 3 black and 3 white balls. also the black balls are the same. and all the white balls are the same.

we have to arrange BBBWWW. the total number of ways is 6!/(3!*3!) = 6*5*4/3! = 20 ways.

also total ways to chose the first 6 balls is 2*2*2*2*2*2 = 64.

so the probability of getting 3 white balls and 3 black in the first 6 choices are =

20/64. also the probability to chose a black ball on the 7th choise is 1/2.

so the answer is 20/64 * 1/2 = 10/64 = 5/32.

this is an example of conditional probability.