Hey wahooza,

Good question - it comes down to the definition of probability, which is that:

The probability of an outcome is equal to the number of times that that outcome occurs divided by the total number of outcomes.

Or, P(A) = A outcomes / Total outcomes


What you're seeing here with the combinations formula is that they're taking the number of combinations of 3 black balls and dividing over the total number of combinations. That works out to:

All-black combinations = 5 black balls, choose 3 = 5!/(3!*2!) = 10

All combinations = 9 balls, choose 3 = 9!/(6!*3!) = 84

10/84 = 5/42




Now, I'd probably do this one differently as I think that it's easier to just look at the sequence of drawing black three straight times. The first time you have:

5 black / 9 total

the second time you have:

4 black / 8 total

And the third time you have:

3 black / 7 total

So you'd multiply:

5/9 * 4/8 * 3/7

And you can simplify that to:

5/9 * 1/2 * 3/7

And then divide the 3 by the 9 to get 1/3:

5/3 * 1/2 * 1/7

Which gets you 5/42


It's the same answer, just two different ways of getting there.


I hope that helps...