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║ ███ ▄ ███ ║  CHAPTER 2
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║ ▄▄▄▄▄▄▄▄▄ ║  PROBABILITY FUNCTIONS
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  A rich variety of the commonly used probability functions are
available for use right on screen.  They provide quick and easy
solutions to the computation of various probabilities that are
often very difficult to perform by hand or by use of a pocket
calculator.  The computations are rapid and very accurate.  All
you need to do is enter a few items of information and you will
quickly have the answers you need.


                           FACTORIALS

  When you choose the factorials option you will then need to
enter the number for which you wish the factorial.  For example,
if you enter the value of N = 12, the program will provide you
with a result of N! = 479001600.  Here are a few more examples.

     N =    8
     Factorial = N! = 40320

     N =   32
     Factorial = N! = 2.631308369336935e+35

     N =  128
     Factorial = N! = 3.856204823625802e+215


                  PERMUTATIONS AND COMBINATIONS

  By choosing the option to compute permutations and
combinations, you will easily obtain both.  You need only enter
the values of n and r to obtain the number of permutations and
the number of combinations of n things taken r at a time.  Here
are some examples that you might want to try for yourself.

     n =   78        r =   13
     Permutations = 1.373031150116532e+24
     Combinations = 220495674290430

     n =   30        r =   15
     Permutations = 2.028432049317273e+20
     Combinations = 155117520


                     BINOMIAL PROBABILITIES

  Binomial probabilities are easily obtained by choosing that
option from the Probability Menu.  You will need to enter the
value of n, your sample size, and the value of r which represents
the number of successes in the sample.  Then enter the value of p
which is the probability of a single successful outcome.  

  Suppose, for example, that you were to throw 30 pennies
(nickels, dimes, quarters or whatever) high up into the air.
When they land, you want to know the probability that exactly
five will show heads up.  In this example, n = 30, r = 5, and
since the probability of a success for a single coin flip is .5,
the value of p = .5.  In this example, the probability of
obtaining exactly five heads is .00013 while the probability of
obtaining five OR MORE heads is .99987.  The program also reports
the mean and variance of the binomial distribution, and in this
example we find that the Mean = 15 and the Variance = 7.5.  Here
is one more example.

     n = 56           r = 11          
     p = 0.3000      
     p( x ) = 0.04538838  
     p( x+) = 0.95461162  
     Mean = 16.80000000 
     Variance = 11.76000000 


                    CHI-SQUARE PROBABILITIES

  Once you know the value of a Chi-square statistic and the
degrees of freedom associated with it, it is a simple matter to
obtain the probability for that Chi-square value.  Merely choose
the Chi-square option in the Probability Menu and then enter the
degrees of freedom and the value of Chi-square.  The following
are two sample results for the Chi-square procedure.

     df = 1    Chi-square = 3.8600       
     p( r ) <= 0.04945     
     df = 4    Chi-square = 6.7800       
     p( r ) <= 0.14798     


                NORMAL CURVE AND t PROBABILITIES

  Probabilities associated with the normal curve and the
t-distribution are combined into a single option that you may
choose from the Probability Menu.  When you choose that option
you need only enter the degrees of freedom for the t-statistic
you have obtained or the sample size if you are justified in
using the normal curve (in that case, df = N).  Then enter the t-
or z-statistic to obtain the probability results.  The program
automatically gives you the area above a positive value of t, the
area below a positive value of t, the area below -t and above +t
(i.e., the area "beyond" -t & +t), and the area from the mean to
the t-value you entered.  The following are sample results using
first a small and then a large sample.

     df = 11
     t or z = 3.860000
     Area above +t       = 0.00133     
     Area below +t       = 0.99867     
     Area beyond -t & +t = 0.00265     
     Area from -t to +t  = 0.99735     
     Area from mean to t = 0.49867     

     df = 289
     t or z = 1.960000
     Area above +t       = 0.02500     
     Area below +t       = 0.97500     
     Area beyond -t & +t = 0.05000     
     Area from -t to +t  = 0.95000     
     Area from mean to t = 0.47500     


              PROBABILITIES FOR THE F DISTRIBUTION

  Probability statistics are easily obtained for the ordinary
F-distribution by choosing that option from the Probability Menu.
Once you choose that option you will need to enter the degrees of
freedom for the numerator (dfn) of your F-ratio, the degrees of
freedom for the denominator (dfd) of your F-ratio, and then the
F-ratio itself.  The following are examples which you may wish to
try on your system.

     dfn = 1            dfd = 11          
     F-ratio = 8.7200      
     Probability <= 0.01314 

     dfn = 3            dfd = 178         
     F-ratio = 2.5600      
     Probability <= 0.05654


                      POISSON PROBABILITIES

  In this procedure Poisson probabilities are always obtained by
entering the Mean, m, of the distribution and a specific value,
x, that is randomly sampled from the distribution.  The procedure
then reports to you the probability of obtaining the value of x
as a random draw and the probability of obtaining a value that
large OR LARGER.  Although you know the value of the Mean and
Variance (the variance of a Poisson distribution is always equal
to the mean), both are reported routinely.  The following are
output examples of the Poisson probability function.

     m = 35.0000     x = 11             
     p( x ) = 0.00000153
     p( x+) = 0.99999934
     Mean = 35.00000000
     Variance = 35.00000000

     m = 35.0000     x = 41             
     p( x ) = 0.03819918
     p( x+) = 0.17506195
     Mean = 35.00000000
     Variance = 35.00000000


                     GEOMETRIC PROBABILITIES

  Geometric probabilities are quickly obtained by entering the
sample size and the probability of a single successful outcome.
For example, if you enter n = 11 and p = 0.5, you will obtain the
results shown below.  A second example is provided for you to try
on your system.

     n = 11           p = 0.50000     
     p( r ) = 0.00024414  
     p( r+) = 0.49951172  
     Mean = 2.00000000  
     Variance = 2.00000000  

     n = 31           p = 0.30000     
     p( r ) = 0.00000473  
     p( r+) = 0.69998422  
     Mean = 3.33333333  
     Variance = 7.77777778  


                  HYPERGEOMETRIC PROBABILITIES

  Hypergeometric probabilities require a wee more input.  First,
you must enter the population size, N.  You must then enter the
number of success, r, in the population.  The number of cases in
your sample, n, is then entered, and you must finally enter x
which is the number of successes in your sample.  The following
are examples which you may want to try for yourself.

     N   = 89               Size of population
     r   = 41               Successes in the population
     n   = 28               Size of sample drawn from N
     x   = 7                Number of successes found in n
     p( x ) = 0.00470293    Prob of x successes
     p( x+) = 0.99859423    Prob of x OR MORE successes
     Mean = 12.8989     
     Variance = 4.8223      

     N   = 114         
     r   = 47          
     n   = 87          
     x   = 21          
     p( x ) = 7.58823665e-12           
     p( x+) = 1.00000000  
     Mean = 35.8684     
     Variance = 5.0369      

     N   = 88          
     r   = 21          
     n   = 34          
     x   = 8           
     p( x ) = 0.20212660  
     p( x+) = 0.61977049  
     Mean = 8.1136      
     Variance = 3.8343      


                    EXPONENTIAL PROBABILITIES

  In this procedure exponential probabilities are obtained by
entering the Mean, m, of the distribution and a specific value,
x, that is randomly sampled from the distribution.  The procedure
then reports to you the probability of obtaining a value that is
larger or smaller than x as a random draw.  Although you know the
value of the Mean and Variance (the variance of an exponential
distribution is always equal to the mean), both are reported
routinely.  The following are output examples of the exponential
probability function.

     Mean = 27.0000      x = 13.0000     
     Area above x = 0.61786735  
     Area below x = 0.38213265  
     Mean = 27.00000000 
     Variance = 27.00000000 

     Mean = 37.0000      x = 41.0000     
     Area above x = 0.33018304  
     Area below x = 0.66981696  
     Mean = 37.00000000 
     Variance = 37.00000000 


                         END OF CHAPTER
