How to calculate the odds of marble picking? Imagine you have 9 marbles in a bag, 2 of them red, and 7 of them black. You pick a marble 3 times, replacing the marble after every pick. What is the probability that you pick exactly 2 red marbles? Is there a formula that can be used to calculate this for any amount of marbles and “picking amount”?
The answer I got, tho I’m not sure if it's right, is 84/729. My logic is that you have a 4/81 chance of picking 2 red with 2 picks total, multiplied by the 7/9 chance that you pick a black marble next time (28/729). You also multiply by the 3 permutations that this can happen in giving an answer of 84/729. Is my reasoning correct?
 A: We want to calculate $$\frac{\text{number of ways we get exactly two red marbles}}{\text{number of ways we get any three marbles}}.$$
Consider the first two times you pick a marble: you have $2$ ways to pick the red marbles, and $9$ ways to pick any marble. Now for the third pick, we do not want a red marble; we then have $7$ choices of black marbles. Therefore, we have
$$\frac{2}{9} \cdot \frac{2}{9} \cdot \frac{7}{9} = \frac{28}{729}.$$ Now we have calculated the probability that the first two marbles are red, and the last marble is black. That is, $P(RRB)$. Of course, there are a total of $3!$ ways to make the drawings, but we consider a drawing of red marble symmetrical, so there are actually $3!/2 = 3$ total such probabilities.
Therefore, the answer is $$3 \cdot \frac{28}{729} = \frac{28}{243},$$ so indeed your answer is correct.
A: It's sad that so many people focus on "formulas" rather than THINKING!
There are 2 red and 7 black marbles.  Since the marbles are returned the probability a red marble is chosen any one time is 2/9 and the probability a black marble is chosen 7/9.
If you choose exactly 2 red marbles the third marble must be black.  That can happen as
red, red, black- probability, (2/9)(2/9)(7/9)= 28/729.
red, black, red- probability, (2/9)(7/9)(2/9)= 28/729
black, red, red- probability, (7/9)(2/9)(2/9)= 28/729.
The probability red,red,black in ANY order is 3 times 28/729= 28/243.
