I've been wondering about this since I got in an unresolved argument with my math teacher in school about 15 years ago, and I'm not sure what made me think about it again today, but I wonder if anyone here can give me a satisfying explanation please...
The question is: you have a bag that contains 10 marbles. 5 are white and 5 are black. If you pull out 3 marbles without replacement, what are the odds that all 3 will be white?
The textbook answer is 5/10 * 5/10 * 5/10 = 1/8
I get how that works for combinations/permutations.
However, that doesn't seem right to me, because I'm pulling these marbles WITHOUT REPLACEMENT.
It makes sense to me if I pull a marble out, it has a 5/10 chance of being white... if I throw it back into the bag and pull another marble, then it also has a 5/10 chance of being white again... but I'm not putting it back! I'm pulling these marbles WITHOUT REPLACEMENT.
So only the first one is a 5/10 chance of being white... When I go to pull the second marble, there are only 9 remaining in the bag, and my odds of the second one being white are 4/9 (not 5/10). Now that I've pulled 2 whites, there are 3 white + 5 black = 8 marbles remaining in the bag, so my odds of the third one being white are 3/8.
So my answer to this homework question was: 5/10 * 4/9 * 3/8
My math teacher said I was wrong and the answer is: 5/10 * 5/10 * 5/10