How can I prove or disprove this for conditional probability?

Question

P(E|S) >= P(E) - P(S)

I assumed it to be true, and started off like this, but am stuck.

P(E) - P(S)

= P(E) - (P(E ∩ S) / P(S))

Not sure how to proceed from here, or if I am even correct in assuming the given statement is true. Please help.

Answer 1

Take two disjoint events $E,S$ with $P(E) >P(S)$. Then $P(E|S)=0 <P(E)-P(S)$. So the inequality is false.

Specific example: on $(0,1)$ with Lebesgue measure take $E=(0,\frac 1 2)$ and $F=(\frac 3 4,1)$.