Great question - and I love that you're asking "why is that the case" instead of just accepting it as a fluke anomaly or something like that...it's in that asking "why" that you tend to really build the kind of flexible knowledge that is such a benefit on this test.
If you wanted to calculate the sequences that would get you to a "third pick = apple" outcome, you can do that:
A, B, A = 2/5 * 3/4 * 1/3 = 1/10
B, B, A = 3/5 * 2/4 * 2/3 = 1/5
B, A, A = 3/5 * 2/4 * 1/3 = 1/10
Sum those up and you get 2/5, which like you said is the same as the initial probability. Here's at least one angle on why:
The "third draw" should have the same probability as the "third position" in a line, right? Whether you're drawing three pieces of fruit without replacement or just lining them up the probabilities shouldn't change. For example, if you were arranging the fruit on a table like:
Apple, Banana, Apple, Banana, Banana
or
Banana, Banana, Apple, Apple, Banana
The probability of having an apple in the third spot should be the same as the probability of drawing an apple third if you were picking them out of a bag, right? And in the lined-up-on-a-table case, it should stand to reason that the probability of an apple in any one position is 2/5 - no one position should have a different probability just because it's in a different spot.
So that's at least one demonstration of why the probability would be the same. The third "position" shouldn't have a different probability than the third "draw", and since that's the case it may be easier to calculate this one using the position model.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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