Probability question: why is $P(X \leq 1, 3 <Y \leq 4) = P(X \leq 1, Y \leq 4) - P(X \leq 1, Y \leq 3)$

Question

Let $X,Y$ be random variables. Can someone explain why

$P(X \leq 1, 3 <Y \leq 4) = P(X \leq 1, Y \leq 4) - P(X \leq 1, Y \leq 3)$

Intuitively, it makes sense, but I'm struggling to prove it with set theory and probability axioms.

Answer 1

This is because we have the set identity $$\{x \leq 1, 3 < y \leq 4\} = \{x \leq 1, y \leq 4\} \setminus \{x \leq 1, y \leq 3\}.$$

EDIT: where we write $A \setminus B$ means $A \cap B^c$ for $B \subseteq A$.

Answer 2

Because $(-\infty,3 ]\times (3,4]$ and $(-\infty ,3]\times (-\infty ,3]$ are disjoints.

Answer 3

Note that $$ A=\{(x,y):(0\le x \le 1, 3\le y \le 4)$$

is a rectangle bounded by $x=1$,$y=3$, and $y=4$

$$ B=\{(x,y):(0\le x \le 1, 0\le y \le 4)$$

is a rectangle bounded by $x=1$,$y=0$, and $y=4$

$$ C=\{(x,y):(x \le 1, 0\le y < 3)$$

is a rectangle bounded by $x=1$,$y=0$, and $y=3$

As you see $$ B=A\cup C$$ and $$A\cap C = \phi $$

Thus $$P(X \le 1, 3 <Y \le 4) = P(X \le 1, 4 \le Y) - P(X \le 1, 3 \le Y)$$

Answer 4

Using only $\cap$ and disjoint sets:

$P(X \leq 1 \cap Y \leq 4) = P((X \leq 1 \cap 3 <Y \leq 4) \cap (X \leq 1 \cap Y \leq 3) ) $

The 2 sets are disjoints so this is equal to:

$= P(X \leq 1 \cap 3 <Y \leq 4) + P(X \leq 1 \cap Y \leq 3 )$

Therefore

$$P(X \leq 1 \cap 3 <Y \leq 4) = P(X \leq 1 \cap Y \leq 4) - P(X \leq 1 \cap Y \leq 3 )$$