# Probability of union

#### hedi

Hi,
LET Ai,Bi i=1,2 be sets such that the A's are independent and the B's are independent.
Also pr(Ai)<=pr(Bi),i=1,2

prove that Pr(A1UA2)<=Pr(B1UB2)

#### Debsta

MHF Helper
What is the formula for P(XUY) if X and Y are independent?

#### hedi

I know the formula but how to proceed?

MHF Helper

2 people

#### hedi

I will be glad for a clue to the solution.

#### Plato

MHF Helper
Hi,
LET Ai,Bi i=1,2 be sets such that the A's are independent and the B's are independent.
Also pr(Ai)<=pr(Bi),i=1,2
prove that Pr(A1UA2)<=Pr(B1UB2)
I know the formula but how to proceed?
You were asked, even politely, to respond to a very simple request. Why the he** did you not? Or is it that you cannot?
I will be glad for a clue to the solution.

1 person

#### hedi

I am stuck with proving that x+y-xy<=u+v-uv when x<=uand
y<=v in (0,1).It is probably simple but somehow i dont come to a solution

#### Plato

MHF Helper
I am stuck with proving that x+y-xy<=u+v-uv when x<=uand
y<=v in (0,1).It is probably simple but somehow i dont come to a solution
Thank you for finally replying. Now look at this very nice solution to a very tricky inequally.

If $\large 0\le a\le x\le 1~\&~0\le b\le y\le 1$ then $\large a+b-a\cdot b\le x+y-x\cdot y$
$$\displaystyle 1-a\geq 1-x \geq 0$$

$$\displaystyle 1-b\geq 1-y\geq 0$$

Multiply

$$\displaystyle (1-a)(1-b)\geq (1-x)(1-y)$$
Multiply out both sides. Then multiply through by $-1$.

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