# Multiplying two event with probability density function, is it possible?

in my exercise, $$X$$ is the size of a tree trunk, and $$X$$ follows a normal distribution $$\mathcal{N}(9,0.4)$$, we want to know $$P(8.8\le X\le11.2)$$

So I though that I could do this: $$P(8.8\le X \le11.2) = P(X\ge 8.8) \cdot P(X\le11.2)$$ since $$P(A \cap B) = P(A)\cdot P(B)$$, but it seems incorrect and I don't get it.

The teacher says we should use $$P(X\le 11.2)-P(X\le 8.8)$$. Does someone know why?

$$P(A\cap B) = P(A) \cdot P(B)$$ works when $$A$$ and $$B$$ are independent, which is not the case here, as $$A$$ and $$B$$ derive from the same observation of $$X$$.
Letting $$f$$ be the probability density function, you have:
\begin{align} P(a\le X\le b) &= \int_a^b f(x)dx\\ &= \int_{-\infty}^b f(x)dx - \int_{-\infty}^a f(x)dx \\&= P(X\le b) - P(X\le a) \end{align}
This can be interpreted as: "if $$X$$ is to be bounded by $$a$$ and $$b$$, it has to be lower than $$b$$ i.e. $$P(X\le b)$$ but not lower than $$a$$ i.e. $$P(X\le a)$$ .
Please note that this does not require $$X$$ to follow a normal distribution.