A 2nd year stat/probability question about P(A|B) I'm sooo close

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A 2nd year stat/probability question about P(A|B) I'm sooo close!

Homework Statement


Consider 2 events A and B such that P(A)=x and P(B)=y, given x,y >0 and x + y > 1

Prove that
P(A|B) >= 1 - [(1-x)/y]


Homework Equations



P(A|B) = P(A n B) / P(B)


The Attempt at a Solution



Now I know that P(A)= x so P(not a) = 1-x and P(B) = y and P(Not B) = 1-y

so the answer that I have to prove is actually the same as 1 - P(not a)/P(b)

So all I have to work out is how to go from

P(A|B) = P(A n B) / P(B)

to

1 - P(not a)/P(b)

And I'm going off the part where x + y > 1

What step am I missing? Am I using the right formula? There must be a trick or formula I can manipulate as this is a question at the start of a chapter so should be easyish..?
 
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laura_a said:

Homework Statement


Consider 2 events A and B such that P(A)=x and P(B)=y, given x,y >0 and x + y > 1

Prove that
P(A|B) >= 1 - [(1-x)/y]


Homework Equations



P(A|B) = P(A n B) / P(B)


The Attempt at a Solution



Now I know that P(A)= x so P(not a) = 1-x and P(B) = y and P(Not B) = 1-y

so the answer that I have to prove is actually the same as 1 - P(not a)/P(b)

So all I have to work out is how to go from

P(A|B) = P(A n B) / P(B)

to

1 - P(not a)/P(b)

And I'm going off the part where x + y > 1

What step am I missing? Am I using the right formula? There must be a trick or formula I can manipulate as this is a question at the start of a chapter so should be easyish..?

1\geq P(A\cup B)=P(A)+P(B)-P(A\cap B)

1\geq x+y-P(A\cap B)

P(A\cap B)\geq x+y-1

\frac{P(A\cap B)}{P(B)}\geq \frac{x+y-1}{y}

P(A|B)\geq \frac{y-(1-x)}{y}

P(A|B)\geq 1-\frac{1-x}{y}
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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