/*-------------------------------------------------------------

  Homwework project #2
  Due Thurs Jan 21, 1993
  This file is /pub/st445/hw2

   We will create our own binomial tables.  You may
recall that in n independent trials with probability
p of success on each trial, the probability of getting
r successes in the n trials is
   Pr(X=r) = n!/[r!(n-r)!]*p**r*(1-p)**(n-r)
so if I have the probability for r-1, I can get the
one for r by multiplying it by (n-r+1)/(r)*p/(1-p).

   Create the binomial tables for n=15 and p = .2, .5,
and .7.  Here is a hint :

   DATA BINOM;
     INPUT R @@;   * @@ keeps us on the same line for
                     reading data ;
     RETAIN;       * RETAIN keeps the values as is for
                     the next pass through the data
                     instead of resetting them to missing.
     N=    ;   P =     ;
     IF R= 0 THEN PROB = (1-p)**n;
     IF R>0 THEN PROB=PROB*                     ;
     CARDS;
   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
     ;

Tasks
  1. Graph the binomial with 2 appropriate title lines.
     Use a LABEL statement to put
       "Probability of R Successes" on the Y axis and
       "Number of Successes = R" on the X axis.
     do a plot for each P and use a footnote to tell what
     P is used on each.
  2. The normal is supposed to be a good approximation
     to the binomial.  What is the appropriate
     mean_______________ and variance ____________
     for each p?
  3. Create R*PROB and R*R*PROB in the data.  Print
     these and sum them using the SUM option of
     PROC PRINT.  Explain how this relates to
     problem 2. Print ONLY the two new variables
     here and add appropriate titles and footnotes.

  4. A child plays a video game at the arcade which
     consists of shooting at 15 targets.  It costs
     25 cents to play and the probability of hitting
     any one target is p and the shots are independent.
     He gets one ticket just for playing (0 hits) and
     the number of tickets doubles for each hit (he
     would get 8 tickets for 3 hits, for example).
     Tickets are taken to a booth for redemption.
     Find the mean number of tickets issued per play,
     that is, per quarter dropped into the machine.
  5. (continuing)  As the arcade owner, I do not want
     to loose money. When I redeem the tickets, how
     much value should I give back per ticket in
     order to make 5 cents on each quarter dropped
     into the machine?  Calculate this for the three
     skill levels .2, .5, and .7 probability of hitting
     any one target.
  6. (continuing) For your suggested ticket redemption value,
     what is the probability of a player in the
     high skill category getting more than 25 cents
     worth of tickets in a run?

         For  questions 7-10 just use p=.2 if you like.

  7. Returning to part 2, graph the normal as * and binomial
     as B overlaid on the same plot.  You'll need
     to compute the normal ordinate f(Z) then use
     the OVERLAY option of PROC PLOT e.g.
       PROC PLOT; PLOT Y1*x Y2*X/overlay;
  8. Plot the deviations of the true binomial
     probabilities from these normal curve
     values against R.
  9. THe so-called normal approximation to the binomial
     involves computing the difference between
         PROBNORM( (R+.5-MU)/SIGMA )
     and
         PROBNORM( (R-.5-MU)/SIGMA )
     where PROBNORM is a function in SAS that calculates
     areas under the normal distribution.
     Calculate the approximation for each R.
        DO these sum to 1?  Enclose a PROC PRINT to
                              show it.
        If not, explain why they don't.
        Plot the difference between the binomial
           probability and the approximation versus R.
           Use appropriate titles labels and footnotes.
  10. A power plant can be fined no more than 15 times
      per year based on 15 inspections for violations
      of the clean air act.  The first fine is $5000, the
      second $10,000 the third $20,000 the fourth $40,000
      etc.   If a power plant has probability .2 of
      being fined on any given inspection, find the
      mean and variance of their yearly fine-paying
      expenses and find the most probable fine and its
      probability.
        What do you think about using
           FINE = 5000*(2**(R-1))*(R>0)   ?
        What is the R>0 doing?
                 */