<PRE><B>Question 482339</B>
your venn diagram should look like this.
&lt;img src = &quot;http://theo.x10hosting.com/problems/venn_diagram_3.jpg&quot; alt = &quot;$$$$$&quot;/ &gt;


Answers to your questions are shown below:


n(A) = the number of elements in A.
That would be 20 + 70 = 90.
20 are in A only.
70 are in both A and B.


n(B) = the number of elements in B.
That would be 50 + 70 = 120.
50 are in B only.
70 are in both B and A.


P(A) = probability of an element shown being in set A.
There are 20 + 70 + 50 = 140 elements in the universe shown.
That universe is A union B.
A union B is the set of all elements that are either in A, or in B, or in both.
Since there are 90 elements in A and there are 140 elements overall, the probability of an element being in set A would be 90 / 140.


P(B) = probability of an element shown being in set B.
There are 20 + 70 + 50 elements in the universe shown.
That universe is A union B.
A union B is the set of all elements that are either in A, or in B, or in both.
Since there are 120 elements in B and there are 140 elements overall, the probability of an element being in set B would be 120 / 140.



P(A|B) = the probability of an element being in A given that it is in B.  
Since B contains 120 elements, and the elements that are also in A are 70, then the probability of an element being in A given that it is in B would be 70 / 120.


P(B|A) = the probability of an element being in B given that it is in A.
Since A contains 90 elements, and the elements that are also in B are 70, then the probability of an element being in B given that it is in  A would be 70 / 90.


Note that the formula for the number of elements in A union B is the elements in A plus the elements in B minus the elements that are in both A and B.
The number of elements in A is equal to 90.
The number of elements in B is equal to 120.
The number of elements in A intersect B is equal to 70.
The number of elements in  A union B are therefore 90 + 120 - 70 = 140.
Your Venn Diagram already shows that relationship.
The number 20 on the left are the elements that are in A only.
The number 50 on the right are the elements that are in B only.
The 70 in the middle are the elements that are in both A and B.
Add them up and you get 140 total elements.
The number of elements in A are 20 + 70.
The number of elements in B are 50 + 70.
You can see the 70 are being double counted because they are the same 70 shown in the middle of the Venn Diagram.
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