Stat I Prof Vinod Binomial Quiz Preparation

   GM wants to test color preferences among car buyers.  It knows that
 50% prefer blue color.  Fifteen buyers were randomly chosen. What is
 the probability that less than a fifth of them will prefer blue?

   Answer:  x= 0,1,2, ...,  15(=n)  is Binomial r.v.
            Parameters of Binom are n and p : here n=15, p=0.5
            one-fifth of n is (15/5)=3
           Less than one-fifth is less than 3
           0 is less than 3 so x=0 is ASKED to be evaluated
           1 is less than 3 so x=1 is ASKED to be evaluated
           2 is less than 3 so x=2 is ASKED to be evaluated
           3 is NOT less than 3 it is equal to 3 
                    so x=0 is NOT ASKED to be evaluated
   So we add P(x=0)= 0.0000
             P(x=1)= 0.0005
             P(x=2)= 0.0032
    Ans is 0.0037 which is their sum
Make a table with headings 
 I   x  Asked? nCx  p^x  n-x q^(n-x)  Product nCxp^xq^(n-x) 
(1) (2)   (3)   (4)  (5)  (6)   (7)    (4)*(5)*(7)


     15 C 0  is 1,  this goes in column (4) along row for i=1 and x=0
     15 C 1  is  15,  this goes in column (4) along row for i=2 and x=1
     15 C 2 is  105,  this goes in column (4) along row for i=3 and x=2

This example is easy because p =q=0.5,  so that p^x *q^(n-x) is simply (0.5)^n
     0.5 to the power 15 is  0.000030517  keep this in memory
     This times 1 is the first number
      This times 15 is the next number
     This times 105  0.003204  etc
p=small p = prob. of one success in one trial, n= largest no of successes.


The answer is simply the sum of items in the last col along 
                                the "Asked" rows in  2nd col.

Sometimes it is more convenient to compute what is NOT ASKED and use
the formula  P(ASKED) = 1 - P(NOT ASKED).
  


Digression:
What is (n C r)?  It is the number of ways of choosing r items from n items
Let n=3 items marked A B and C
r=2.   How many ways can you select 2 items from 3 items?
     Answer:   AB, AC and BC.  this means in 3 ways

n C r = Numerator / Denominator,
   Numerator= n! = 1x2x3x.....xn= multiplication of all these numbers
   Denominator = r!(n-r)!
   In our example, n=3 and r=2,  n!=1x2x3=6,  r!=1x2=2 and (n-r)!=(3-2)!=1!=1
   so the numerator =6
          denominator =2,  ratio = 6/2 =3,  htese are the 3 ways enumerated 
          above.  AB, BC and AC