The way I solved this was (1/5)^6, however that answer came out to be 6.4e-5. However the answer is suppose to be 0.0003200. Can I get a little help on what I did wrong? Tt
You found the probability that all six witnesses picked a particular person, not that they all picked the same person. So, let the first witness pick any of the five, then find the probability that the next five witnesses pick that same person.
After the first witnesses pick a person, each of the remaining 5 witnesses must pick that person, each with probability . So . There's no restriction on who the first person is, just the remaining five must pick the same one.
Edit - sorry, I misread the above post. Yeah what Mark said is correct.
Another way to see this is that the probability of all 6 witnesses picking a particular man is . But there are ways to choose a particular man from all 5 men. Therefore the probability of all 6 witness choosing the same man is .
Yet another completely equivalent way to see the answer is to think of the space of possibilities, which is a set of all the ordered choices of the 6 witnesses:
Let = set of possibilities = , where .
So, for instance, the element would correspond to witness #1 choosing man #3,
witness #2 choosing man #5, witness #3 choosing man #2, and so forth. It's one possibility of the choices of the 6 witnesses.
Let be the set of elements of which represent "all witnesses chose the same man". Then
By the definition of probability, the probability that "all chose the same man" = .