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

Question

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?

Answer 1

$P(A\cap B) = P(A) \cdot P(B)$ 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.