Question 73262
<pre><font size = 4><b>There are 8 disjoint regions:
A&#8745;B&#8745;C, A&#8745;B&#8745;C', A&#8745;B'&#8745;C, A&#8745;B'&#8745;C', A'&#8745;B&#8745;C, A'&#8745;B&#8745;C', A'&#8745;B'&#8745;C, A'&#8745;B'&#8745;C'. 

The "sieve" formula for 3 sets:

n(AUBUC) = n(A)+n(B)+n(C)-n(A&#8745;B)-n(A&#8745;C)-n(B&#8745;C)+n(A&#8745;B&#8745;C)

[If you haven't had this sieve formula, please post again, or email
me AnlytcPhil@aol.com and I can show you how to do it without that
formula.  But it's easier if you have it.]

We also know

n(A&#8745;B&#8745;C) = n(U) - n[(A&#8745;B&#8745;C)'] = 60 - 13 = 47

Substituting what we have in the sieve formula:

47 = 21 + 30 + 30 - 11 - 13 - 17 + n(A&#8745;B&#8745;C)

47 = 40 + n(A&#8745;B&#8745;C)

n(A&#8745;B&#8745;C) = 7  

From that we can get the number of elements in each of the other
seven disjoint regions:

1. n(A&#8745;B&#8745;C) = 7
2. n(A&#8745;B&#8745;C') = n(A&#8745;B)-n(A&#8745;B&#8745;C) = 11-7 = 4 
3. n(A&#8745;B'&#8745;C) = n(A&#8745;C)-n(A&#8745;B&#8745;C) = 13-7 = 6
4. n(A'&#8745;B&#8745;C) = n(B&#8745;C)-n(A&#8745;B&#8745;C) = 17-7 = 10
5. n(A&#8745;B'&#8745;C') = n(A)-n(A&#8745;B&#8745;C')-n(A&#8745;B'&#8745;C)-n(A&#8745;B&#8745;C) = 21-4-6-7 = 4
6. n(A'&#8745;B&#8745;C') = n(B)-n(A&#8745;B&#8745;C')-n(A'&#8745;B&#8745;C)-n(A&#8745;B&#8745;C) = 30-4-10-7 = 9
7. n(A'&#8745;B'&#8745;C) = n(C)-n(A&#8745;B'&#8745;C)-n(A'&#8745;B&#8745;C)-n(A&#8745;B&#8745;C) = 30-6-10-7 = 7

a) P(A&#8745;B&#8745;C)

n(A&#8745;B&#8745;C)=47, so P(A&#8745;B&#8745;C) = 47/n(U) = 47/60 

----------------------

b) P(AUB)

n(AUB)=n(AUBUC)-n(A'B'&#8745;C) = 47-7 = 40, so

P(AUB) = n(AUB)/n(U) = 40/60 = 2/3

-------------------------

c) P[(A U C)'] 

n(AUC) = n(AuBuC)-n(A'&#8745;B&#8745;C') = 47 - 9 = 38, so

P(AUC) = 38/40 = 19/20, and

P[(AUC)'] = 1 - P(AUC) = 1 - 19/20 = 1/20
 
------------------------

d) P(A&#8745;B&#8745;C|A U B U C) =

 P[(A&#8745;B&#8745;C)n(AUBUC)]      P(A&#8745;B&#8745;C)     n(A&#8745;B&#8745;C)/n(U)     7/60 
-------------------- = ----------- = --------------- = ------- = 7/47
     P(AUBUC)            P(AuBuC)     n(AuBuC)/n(U)     47/60


------------------------

                  P(An(BUC)')                            P(A&#8745;B'&#8745;C')
e) P[A|(BUC)'] = ------------- = (by DeMorgan's law) = ------------- = 
                   P[(BUC)']                              P(B'&#8745;C')

   n(A&#8745;B'&#8745;C')/n(U)        4/60            4
  ----------------- = ------------- = ----------
     P(B'&#8745;C')          n(B'&#8745;C')/60     n(B'&#8745;C')

Since n(B'&#8745;C') = n(A&#8745;B'&#8745;C')+n(A'&#8745;B'&#8745;C') = 4+13 = 17, the above = 4/17

------------------------

               P(A'&#8745;B')                           P[(AUB)']
f) P(A'|B') = ---------- = (by DeMorgan's law) = ----------- =  
                 P(B')                              P(B')

 1 - P(AUB)                        1 - 2/3           1/3
------------ = [By prob (b)] = --------------- = ----------- = 
  1 - P(B)                      1 - n(B)/n(U)     1 - 30/60 

   1/3       1/3 
--------- = ----- = (1/3)(2/1) = 2/3 
 1 - 1/2     1/2 

Edwin</pre></b></font>