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...
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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