To my knowledge, "with replacement" means that rock selected for your first draw is put back in the lot. This makes the second draw independent from the first, as explained here:
https://math.youngzones.org/joint.html
The probability that the first draw is a slate rock is 15/45 (number of favorable cases/number of possible cases) or 1/3. The probability of the second rock being a slate rock will be 1/3 as well, since we have replaced the initial draw in the lot (i.e. we've put it back in the set).
The probability of both events taking place is 1/3 * 1/3 or 1/9.
HOWEVER
If you consider a "without replacement" probability, then you get something different. The first draw will have the same probability, i.e. 1/3. The second draw will not, since the number of favorable cases has dropped from 15 to 14 (since you've drawn a slate rock in your first pick) and the total number of cases has dropped from 45 to 44. So for your second draw, you get probability of = 14/44 = 7/22.
Multiply the two to get 7/66.
What is the source of this problem? Not all sources are 100% error-proof. Neither is my explanation, so I'm waiting for other opinions as well.
