Good problem! I love GMAT probability strategy, and quite a bit of it comes into play here. A few questions you should ask yourself include:

1) Does the probability of the first event change the probability for the second event?

Here, it does. We start with 15 coins and we'll draw one out and not replace it, so the second event only has 14 coins. (More on that in a second)

2) Do you need a particular sequence to happen, or will multiple sequences accomplish your goal?

Here there is only one sequence that we need: Not Nickel, then Not Nickel.

Sometimes, however, there are multiple sequences that would get you what you need. If this question were "what is the probability of selecting exactly one nickel, then two sequences would work: Nickel --> Not Nickel; or Not Nickel -->Nickel

In this case, we need to multiply our one sequence:

Not Nickel, Not Nickel

The first has a probability of 9 non-nickels / 15 total coins
The second has a probability of 8 non-nickels / 14 total coins (you'll assume that you got what you wanted on the first draw, as we're multiplying the "new probability" by the 9/15 chance that we get what we wanted the first time)

Then multiply:

9/15 * 8/14 is:
3/5 * 4/7 = 12/35

Brian@VeritasPrep wrote:Good problem! I love GMAT probability strategy, and quite a bit of it comes into play here. A few questions you should ask yourself include:

1) Does the probability of the first event change the probability for the second event?

Here, it does. We start with 15 coins and we'll draw one out and not replace it, so the second event only has 14 coins. (More on that in a second)

2) Do you need a particular sequence to happen, or will multiple sequences accomplish your goal?

Here there is only one sequence that we need: Not Nickel, then Not Nickel.

Sometimes, however, there are multiple sequences that would get you what you need. If this question were "what is the probability of selecting exactly one nickel, then two sequences would work: Nickel --> Not Nickel; or Not Nickel -->Nickel

In this case, we need to multiply our one sequence:

Not Nickel, Not Nickel

The first has a probability of 9 non-nickels / 15 total coins
The second has a probability of 8 non-nickels / 14 total coins (you'll assume that you got what you wanted on the first draw, as we're multiplying the "new probability" by the 9/15 chance that we get what we wanted the first time)

Then multiply:

9/15 * 8/14 is:
3/5 * 4/7 = 12/35

Thanks Brian,
Nice work