I'm trying to solve a problem two different ways, and I can'd seem to figure out where I'm going wrong.
I have 4 buckets (A,B,C,D), and 4 identical marbles. Each marble has an equal chance of being put in any of the 4 buckets, and each is placed independently (each bucket can have 0-4 marbles placed in it, with a total of 4 across all buckets)
I need to calculate the probability of bucket A being empty after the four marbles are placed.
My intuition says I can calculate the probability of the marble being placed in to any bucket except A as $\frac{3}{4}$. Then I can multiply the probabilities together since they are independent. So I can do $\left(\frac{3}{4}\right)^ 4$, which seems right.
But I think I should also be able to get the answer by calculating the number of ways that leaves A empty divided by the total number of ways the marbles can be placed in the 4 buckets. When I do that I get:
total ways: $\left(\dbinom{4}{4}\right)$ $\longrightarrow$ $\dbinom{7}{4}$ $\longrightarrow$ 35
ways with A empty: $\left(\dbinom{3}{4}\right)$ $\longrightarrow$ $\dbinom{6}{4}$ $\longrightarrow$ 15
But that gives me $\frac{15}{35}$, which is not the same as $\left(\frac{3}{4}\right)^ 4$
I'm guessing I am over counting or something else dumb, but I'm really stumped. Thanks for the help!