What is the probability distribution of X, if X is the number of times the letter 'e' appears from the set {beware, the, jabberwock, my, son}

Question

A string of letter is chose uniformly at random from the set

{beware, the, jabberwock, my, son}

Let X be the number of times the letter 'e' appears in the string. Give the probability distribution of X.

This is how I attempted it:

let 1 represent e and 0 represent not e

X= {010001, 001, 0000100000, 00, 000}

Is this correct?

Answer 1

No, it's not correct. First, here $X$ is a number (random), not a set.

When you're asked "give the probability distribution of $X$", your answer needs to be

$P(X=a) = ?$, $P(X=b) = ?$, etc.

So, what's the probability that $X=0$? that $X=1$ and $X=2$? Is there another value that $X$ can take ?

Answer 2

After seeing some comments I believe this would be my answer

P(X= Beware) = 2/5

P(X= the) = 1/5

P(X = jabberwork) = 1/5

P(X= my) = 0

P(X= son) = 0

Feel free to state whether this is right or wrong