#   Topic 11

     # probability is always between 0 and 1
     #
     # demonstrate a small "catch" to this, namely
     # that R expresses really small, close to 0,
     # values in "scientific" form.  For example,
     # look at the number 0.00000006478.
0.00000006478

     # we will run 4 trials of the experiment of
     # getting 5 random values between 1 and 99.
     # Note that if you run this script then you
     # will get different results than are shown
     # here. Sort the values to make it easier to
     # inspect the lists.
sort( as.integer( runif( 5, 1, 100)) )
sort( as.integer( runif( 5, 1, 100)) )
sort( as.integer( runif( 5, 1, 100)) )
sort( as.integer( runif( 5, 1, 100)) )

     # Set up three different choices
blue_hat <- c(1,2,3,4,5,6,7,8)
coin <- c("H","T")
fruit <- c("O","P","A","G","L")
     # Get thee sample space of an item
     # taken from each choice
expand.grid( blue_hat, coin, fruit )

     # look at permutations
     # we need to load a special package to get
     # the function we want.
install.packages("gtools")
library(gtools)
     # recall the values in fruit
fruit
     # Now that we have the library installed we
     # can use the function permutations to get the
     # permutations of our 5 fruits taken 3 at a time.
permutations( 5, 3, fruit )
     # demonstrate factorial()
factorial(8)
factorial(6)
     # look at the surprising case
factorial( 0)
     # find the number of permutations of 19 things
     # taken 4 at a time
factorial( 19 ) / factorial( 19-4 )
     # load all of the related functions into the
     # environment
source("../combinations.R")
     # compute the answer to the previous problem
     # using the two functions nPr() and num_perm().
nPr( 19, 4)
num_perm( 19, 4 )
     # get the combinations of the 5 fruits taken
     # 3 at a time
combinations( 5, 3, fruit)
     # find the number of combinations of 17 things
     # taken 6 at a time
factorial( 17 ) /(factorial(17-6)*factorial(6))
     # do this with the functions nCr and num_comb
nCr( 17, 6 )
num_comb( 17, 6 )