Finding probabilities from a piecewise cdf function

Question

This is the question

I want to find $P(1\leq x<2)$. The answer key says that it is $1/4$. I am not sure how that is the answer, as I am thinking that it should be $P(x<2) - P(x\leq1)$ which is $3/4 - 2/3 = 1/12$. I'm assuming that the answer is found with $43/4 - 1/2 = 1/4$, but where does that $1/2$ come from? Is it $P(x<1)$?-

Answer 1

$F(x)=P(X \leq x)$. For $P(X<x)$ you have to take the left-hand limit of $F$ at $x$: $P(X<x)=\lim_{y\to x, y<x} F(y)$. So $P(X<2)=\frac 9 {12}$ and $P(X<1)=\frac 1 2$. So the answer is $\frac 9 {12}-\frac 1 2=\frac 1 4$.

Answer 2

The $\frac12$ is indeed $P(X<1)= \lim\limits_{x\to 1^-}F(x) $, as in :

$$\begin{align} & P(1\le X<2) \\= & P(X <2) - P(X \lt 1) \\= & \lim\limits_{x\to 2^-}F(x) -\lim\limits_{x\to 1^-}F(x) \\= & \left(\frac2{12}+\frac7{12}\right)- \frac12 \\= & \frac9{12}- \frac12 \\= & \frac1{4}\end{align}$$

In both cases, you are taking $\lim F(x)$ from below.