Question 430559
There are two sets of marbles.  Both have 1,000 in them.  Bag 1 has 700 blue and 300 red.  Bag 2 has 300 blue and 700 red.  Picking a bag at random you draw out 10 marbles with replacement.  If 7 are red and 3 are blue, what is the exact probability of you picking Bag 1?
<pre>
                      P(bag1 & 7R3B)     
P(bag1 given 7R3B) = ----------------  
                         P(7R3B)

P(7R3B|bag1) = C(10,3)[(.7)^7(.3)^3] 

P(7R3B|bag2) = C(10,3)[(.3)^7(.7)^3] 
 
Numerator = P(bag1 & 7R3B) = P(bag1)*P(7R3B|bag1) = (.5)C(10,3)(.7)^7(.3)^3

Denominator = P(7R3B) = P[(bag1 & 7R3B) OR (bag2 & 7R3B)] = 
             (.5)C(10,3)[(.7)^7(.3)^3] + (.5)C(10,3)[(.3)^7(.7)^3]
  
Numerator = P(Bag2 & 7R3B) = P(Bag2)*P(7R3B|Bag2) = (.5)C(10,3)(.3)^7(.7)^3


              (.5)C(10,3)(.7)^7(.3)^3
-----------------------------------------------------
(.5)C(10,3)[(.7)^7(.3)^3] + (.5)C(10,3)[(.3)^7(.7)^3]


Divide top and bottom by (.5)C(10,3)

        (.7)^7(.3)^3
-----------------------------
 (.7)^7(.3)^3 + (.3)^7(.7)^3

Divide the top and bottom by (.7)^3(.3)^3

        (.7)^4
---------------------------
   (.7)^4 + (.3)^4

Multiply top and bottom by 10^4

         7^4
    -------------
      7^4 + 3^4

        2401
    -------------
      2401 + 81 

        2401
       ------
        2482

Edwin</pre>