I have the following practice problem.
From a deck of five cards numbered 2, 4, 6, 8, and 10, a card is drawn at random and replaced. This is done three times. What is the probability that the card numbered 2 was drawn exactly two times, given that the sum of the numbers on the three draws is 12?
I keep ending up at the following:
$3\cdot\left(\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}\right)=\frac{3}{125}$
However, the provided answer is $\frac{3}{10}$.
My reasoning comes from doing a logic tree where you have $\frac{1}{5}$ probability of choosing 2, followed by $\frac{1}{5}$ probability of choosing 2, followed by $\frac{1}{5}$ probability of choosing 8 (in order to get a sum of 12). There are three permutations so I end up with my solution above.
Where am I going wrong?