Using Tables in Probability Problems -how to make problems easier

Suppose we have a problem like: First draw a table to show A, A c,B, B c, and choose an arbitrary number for the total. 100 and 10,000 are good choices. If A  B = S (sample space), P(A and B c ) = 0.25, and P(A c ) = 0.35, then P(B) =?

Next fill in any obviously easy item, e.g., P(A c )=.35 If the total is 100 and 35% are A c, then 35 are A c.

Now we see that 65 must be the total for A. We are also told that P(A and B c ) = 0.25, so we can fill in 25 for that cell.

Now we see that the number for (A and B) must be 40.

Now if all A and B together occupy the complete sample space ( A  B = S), then the three shaded cells must add to 100. So, B and A c must be 35.

So the total must be 75 for B, and P(B) =.75.