Probability

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fred2028

Junior Member
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Apr 10, 2006
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101
The question provides the following information. != means not equal to

P(A) != 0
P(B) != 0
P(A U B) = P(A) + P(B) - 0.1
P(A|B) = 0.2

I've calculated the following as previous parts of the question

P(AB) = 0.1
P(A'|B) = 0.8
P(B) = 0.5

The current part asks for P(A'). I've stared at this for a bit, and I don't know how to start. Any ideas? Thanks.
 
fred2028 said:
The question provides the following information. != means not equal to

P(A) != 0
P(B) != 0
P(A U B) = P(A) + P(B) - 0.1
P(A|B) = 0.2

I've calculated the following as previous parts of the question

P(AB) = 0.1
P(A'|B) = 0.8
P(B) = 0.5

The current part asks for P(A'). I've stared at this for a bit, and I don't know how to start. Any ideas? Thanks.
Click to expand...
Hmmm. There are four independent areas on an A-B Venn diagram, and we only know two of them. In fact, only two numbers show in the problem. Did a number get lost in transmission? Otherwise I think the best you can do is write an equation P(A')=... where there is an unknown term remaining on the right side.
 
DrPhil said:
Hmmm. There are four independent areas on an A-B Venn diagram, and we only know two of them. In fact, only two numbers show in the problem. Did a number get lost in transmission? Otherwise I think the best you can do is write an equation P(A')=... where there is an unknown term remaining on the right side.
Click to expand...
The sheet says I could give the tightest numerical bounds too, what does that mean?
 
fred2028 said:
The sheet says I could give the tightest numerical bounds too, what does that mean?
Click to expand...
AH! That is the clue.

The total area of (AB') + (AB) + (A'B) + (A'B') = 1

You already know B=0.5 and AB=0.1. What is the smallest/largest that A could be?

A ' =1 - A .. what is the largest/smallest that A' could be?
 
DrPhil said:
AH! That is the clue.

The total area of (AB') + (AB) + (A'B) + (A'B') = 1

You already know B=0.5 and AB=0.1. What is the smallest/largest that A could be?

A ' =1 - A .. what is the largest/smallest that A' could be?
Click to expand...
Wait, in that case, isn't A' just B - AB = 0.4 since A and B overlap and everything that has an A is just invalid?

Also next part is P(A U B U A' U B'). I know that A U B = A + B - AB, but union-ing the A' and B' overlaps with A and B, so am I supposed to double count the probabilities or just ignore them?
 
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fred2028 said:
Wait, in that case, isn't A' just B - AB = 0.4 since A and B overlap and everything that has an A is just invalid?

Also next part is P(A U B U A' U B'). I know that A U B = A + B - AB, but union-ing the A' and B' overlaps with A and B, so am I supposed to double count the probabilities or just ignore them?
Click to expand...
In your first question, B- AB tells you nothing about A. It just tells you those B that are not also A.

The second question can be solved in a simple way or a hard way. Simple way.

The possibilities are AB, AB', B'A, B'B'. That's all there is. So what is the sum of their probabilities?
 
fred2028 said:
Also next part is P(A U B U A' U B'). I know that A U B = A + B - AB, but union-ing the A' and B' overlaps with A and B, so am I supposed to double count the probabilities or just ignore them?
Click to expand...
Any time you take a Union, the cross-term (like AB) is there to subtract away the double possibility, so you don't have to "ignore" it. If you ever get a number bigger than 1, you have done something wrong.

Regroup the question: P = ((A U A') U (B U B'))

What is the union of A and A'? What is the union of B and B'?
 
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