set notation/probability point I cannot resolve

If I have set A = {1,3,5,7,9} and set B = {1,4,9}

Then could I say P(A) = 5/6 and P(B) = 1/2 ???

But if so then P(A intersect B) should be 5/6 * 1/2 = 5/12

But the actual intersection is 2/6 or 1/3. So what is wrong with this reasoning?

In the above the universal set has not been defined? Is that the problem? But can we not assume universal set is 1,3,4,5,7,9? Why does the probability theory that P(A interset B) = P(A) x P(B) not work in this case?

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Originally Posted by angypangy

If I have set A = {1,3,5,7,9} and set B = {1,4,9} Then could I say P(A) = 5/6 and P(B) = 1/2 ???
But if so then P(A intersect B) should be 5/6 * 1/2 = 5/12
But the actual intersection is 2/6 or 1/3. So what is wrong with this reasoning? In the above the universal set has not been defined? Is that the problem? But can we not assume universal set is 1,3,4,5,7,9? Why does the probability theory that P(A interset B) = P(A) x P(B) not work in this case?

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If the universal set has not been defined, then I don't think you can actually do this problem.
If you can assume that U={1,3,4,5,7,9} then P(A^B)=2/6=1/3 because A^B={1,9}.

But I thought probability theory says that P(A ^ B) = P(A) x P(B). But that does not work in this case. Why is that? Is there some pre-requisite for this rule to work that is not valid here?

My difficulty is why the P(A ^ B) = P(A) x P(B) rule does not work here?

Quote:

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Originally Posted by angypangy

My difficulty is why the P(A ^ B) = P(A) x P(B) rule does not work here?

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That does not work in general.
The multiplication rule only holds if the events are independent.