I understand that the formula for calculating a conditional probability is the following $P(A \mid B) = \frac{P(A \cap B)}{P(A)}$
I have this probability to calculate: $P(2\le X \le 3 \mid X \ge1)$.
What I would normally do is $P(2\le X \le 3 \mid X \ge1) = \frac{P(2\le X \le 3 \cap X \ge1)}{P(2\le X \le 3)}$.
However the numerator equals $P(2\le X \le 3)$. The above probability automatically becomes $1$ which is not right.
The textbook says the correct formula is this: $P(2\le X \le 3 \mid X \ge1) = \frac{P(2\le X \le 3 \cap X \ge1)}{P(X \ge1)}$.
Why is that? I thought the denominator is always the first part of the conditional probability like the first formula.
The formula for calculating a conditional probability that you wrote is not correct; you should divide by $P(B)$. See https://en.wikipedia.org/wiki/Conditional_probability