There are six black balls and three white balls in the black box. The probability of getting the black ball for the second time (pay attention to the classification) is easy to handle below.
Method 1:
1. The probability of getting the white ball for the first time is 3/9, and the probability of getting the black ball for the second time is 6/8, so the multiplication of the two is1/4;
2. The probability of getting the black ball for the first time is 6/9, and the probability of getting the black ball for the second time is 5/8, so this multiplication is 5/12;
So the sum of the above two categories is 2/3.
Method 2:
This method is equivalent to the problem of permutation and combination, and it is equivalent to the arrangement when nine people sit in the position and the second seat has been fixed as a black ball. Since the second position is black, taking a black ball from six black balls and putting it on seat 2 means that there are six ways to take it, while the other eight positions have a (8,8) ways to take it, and the denominator is that there is always a (9,9 * *).
6*A(8，8)/A(9，9)=2/3
Besides, your question is not clear. Do you want to ask whether you got the black ball for the third time or after the third time? If the former, I have solved it; If it is the latter, I need to change it.