You are distributing 3 balls among 4 boxes, each one randomly and independent of the others - what is the probability that all 3 balls go to different boxes? - For this problem, what would the answer be? I am thinking:
4C3 / ((3+4-1)C3) = 1/5, where the numerator is the number of configurations with each ball in a different box, and the denominator is all possible configurations of 3 balls in 4 boxes.
Is this question different from: I have a square, and place three dots along the 4 edges at random. What is the probability that the dots lie on distinct edges? - Because the answer to this question from what I have gathered is 1 * 3/4 * 2/4 = 3/8.
Why are the two questions different? Thanks!
Your answer to the first question is wrong: it is similar to the square question and has the same answer of $\frac38$ and can be solved exactly as it has been for the second problem.
However, there is more than one way to solve the first question, eg Place balls in $4*3*2$ ways against a total possible of $4^3$ ways, giving $Pr= \dfrac{24}{64}=\dfrac38$
Note
By the way, your attempt for the first question is quite confused. "You are distributing $3$ balls among $4$ boxes, each one randomly and independent of the others"
(i) You need to be clear whether a problem is a permutation type or combination type or (sometimes) solvable using either of the two in a consistent way. (ii) Stars and bars does not distribute randomly and independently of others