Basic query Related to dependent random variables $X$ and $Y$ are two dependent random variables. I want to find the following probability $$\Pr(2X<c,4Y>c)$$ wher $c$ is some positive number. In my attempt, I can expand the above probability using conditional probability and can get the following result $$\Pr(2X<c\mid 4Y>c)-\Pr(2X<c\mid 4Y>c)\Pr(4Y<c).$$ However in one of the paper that I am studying right now state the result to be $$\Pr(2X<c)-\Pr(2X<c,4Y<c).$$ I do not know how to reach to this result. Please help me in getting to it. Thank you.

The Law of Total Probability, is: $~\mathsf P(A)=\mathsf P(A, B)+ \mathsf P(A, B^\complement)$
In this case $A\equiv \{2X<c\}, B\equiv \{4Y\leqslant c\}, B^\complement\equiv\{4Y>c\}$ so we have:
$$\mathsf P(2X<c) ~=~ \mathsf P(2X<c, 4Y\leqslant c)~+~\mathsf P(2X<c, 4Y>c)$$
Then it is just a matter of algebraic rearrangement:
$$\mathsf P(2X<c, 4Y>c)~=~\mathsf P(2X<c) ~-~ \mathsf P(2X<c, 4Y\leqslant c)$$
That is all.  

Okay, with the addendum that only if $~\mathsf P(4Y=c) = 0~$, then  $$\mathsf P(2X<c, 4Y>c)~=~\mathsf P(2X<c) ~-~ \mathsf P(2X<c, 4Y< c)$$
