Question

I have the solutions here and do not require the answer. Instead, I would like an explanation of why the solutions are the way they are.

a.) This is (bins)^(balls)? Correct? What if there were indistinguishable balls? Would there be another formula?

b.) The answer is (2^5)/(3^5) for the first bin to be empty but wouldn't this be same probability be the same for ANY of the three bins to be empty?

c.) This one I have a good understanding, I think. It's a simple probability like the ball is going in one of three bins 3 * (1 of three bins) 1 / the probability of 3^5. Please confirm.

d.) This one, in particular, is the most confusing to me. Please, and I cannot stress this enough, explain it to me as if I was 5 years old. I saw the solution, but why do you subtract 1 - the information displayed.

e.) This one is a biconditional probability, but please explain the process for this too.

Please, try and explain with clarity and precision as I really want to grasp these concepts.

4. Suppose we throw 5 labelled balls randomly into 3 labelled bins. A. How many outcomes are there in the sample space? B. What is the probability that the first bin is empty? C. what is the probability that at least two bins are empty? D. What is the probability that no bin is empty? E. What is the probability that the first bin is empty, given that the second bin is empty?

Answer 1