Simple probability mass function question

Question

I was wondering why given the probability of an event $P(X>n)=\frac{1}{(n+1)}$ then $P(X=n)=\frac{1}{n(n+1)}$. Could anyone lend me some help please? Thank you so much

Answer 1

Hint: Since $\{n\} \cup \{n+1,n+2,...\} = \{n,n+1,n+2,...\}$, and these sets are disjoint, we have $P \{n\} + P \{n+1,n+2,...\} = P \{n,n+1,n+2,...\}$.

Details:

$$P \{n\} = P \{n,n+1,n+2,...\} - P \{n+1,n+2,...\}= \frac{1}{n} - \frac{1}{n+1}= \frac{1}{n(n+1)}$$

Answer 2

Presumably $N$ and $X$ are the same.

Hint: If $X$ takes only integer values, $X = n$ means: $X > n-1$ but not $X > n$.