Four balls are to be randomly selected, with replacement, from an urn that contains $20$ balls numbered $1$ through $20$. If the random variable $X$ is the largest numbered ball selected, find the following:
$a)$ Probability that $X$ takes on each of its possible values.
$b)$ Probability that $X$ is strictly greater than $10$.
My answers are:
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a) \ \ \ \frac{i^3}{20^4} \ \ \ \ \ \ i=1,2,\ldots,20$$
$$b) \ \ \ \ 1-\frac{10^4}{20^4}$$
Could someone please verify these? I am a little unsure of my answers.
Best Answer
(a) is not correct, you haven't accounted for the ordering of the balls. Try summing the probabilities from $i = 1$ to $i = 20$ and you won't get 1. (b) looks fine. A hint would be if you can work out the probability that $X > n$ for any $n$, you can quite easily work out the probability that $X = n$.