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The sixth ball is white

A box contains 7 identical white balls and 5 identical black balls. They are to be drawn randomly one at a time without replacement until  the box is empty. Find the probability that the 6th ball drawn is white, while before that exactly 3 black balls are drawn.

''Principle and techniques in combinatorics by Chen Chuan Chong, Ch 1, quest. 24''

$$\frac{good} {all} = \frac { \mathit2X \cdot \mathit 3Y }{\mathit 5(X+Y)} \frac {X}{X+Y} \frac {\mathit 4X \cdot \mathit 2 Y} {\mathit 6(X+Y)} $$

Above is the species presentation. The e.g.f's are

$$good = \frac {x^2y^3}{2!.3!} {x} \frac {x^4y^2}{4!.2!}  $$

$$all ={ (x+y)^5 \over 5!} (x+y) { (x+y)^6 \over 6!} $$

Now we compare the coefficients of $$ x^7y^5\over 7!5! $$ in the two e.g.f.'s above and we get:

$$\frac {1050}{5544} = \frac {25} {132} $$