Bigger than and equals rewritten in normal distribution question

Question

So it is correct to say that $P(482\le x \le 510) = P(x \le 510) - P(x < 482)$ where x is a random variable in a normal distribution? Thanks!

Answer 1

In general, $P(A\setminus B)=P(A)-P(B)$ whenever $B\subseteq A$. Now if $a<b$, then $$ P(a\leq X\leq b)=P\big(\{X\leq b\}\setminus \{X<a\}\big)=P(X\leq b)-P(X<a), $$ since $\{X<a\}\subseteq\{X\leq b\}$.