# Some examples of binominal probabilities
#
#  First, introduce pbinom().

#  What is the probability of getting 8 or fewer 
#  successes in 23 trials, if the probability of success
#  is 0.42?
pbinom( 8, 23, 0.42 )
#
#  What is the probability of getting 7 or fewer 
#  successes in 23 trials, if the probability of success
#  is 0.42?
pbinom( 7, 23, 0.42 )
#
#  Then, let us take care of getting an exact 
#  probability.
#
#  What is the probability of getting exactly 8
#  successes in 23 trials if the probability of
#  success is 0.42?
#
pbinom( 8, 23, 0.42) - pbinom( 7, 23, 0.42)
#
#  Alternatively, we do have a special function, 
#  pbinomeq() that will find that same value
#  but we need to load it into our environment.
#
source("../pbinomeq.R")
pbinomeq( 8, 23, 0.42)

#
# Then we start to look at stranger questions.
#
#  what is the probability of getting anything
#  other than 8 successes out of 23 trials when
#  the probability of success is 0.42?
#
1 - pbinomeq( 8, 23, 0.42)
#
#  What is the probability of getting more than
#  8 successes out of 23 trials witen the 
#  probability of success is 0.42?
#
1 - pbinom( 8, 23, 0.42 )
#
#  We should note that R has a special parameter
#  that is often used to change the direction of
#  our view of the probabilty distribution.  That
#  parameter is lower.tail, and we can demonstrate
#  it here.
#
#  The probability of getting 8 or fewer successes
#  for our case was
pbinom( 8, 23, 0.42 )
#  The probability of getting more than 8 is
1 - pbinom( 8, 23, 0.42 )
#  but we could have done that by
pbinom( 8, 23, 0.42, lower.tail=FALSE)
# 
#   Note that pbinom( 8, 23, 0.42 ) gave us
#      8 or less
#  but that pbinom( 8, 23, 0.42, lower.tail=FALSE)
#      gives us more than 8,
#   specifically, not "8 or more'!
#
#  What is the probability of getting fewer
#  than 8 successes in 23 trials with the 
# probability of success = 0.42?
#
#  "fewer than 8" is the same as 
#  # 7 of less" so
pbinom( 7, 23, 0.42)
#
#  We could have done this same problem by saying 
#  "fewer than 8" is the same as the complement
#  of "more than 7"
1 - pbinom( 7, 23, 0.42, lower.tail=FALSE)
#
#  What is the probability of getting between
#  10 and 14 successes out of 23 trials,
#  including 10 and 14, with
#  the probability of a success = 0.42?
#
# pbinom( 14, 23, 0.42) is the probabilty 
# of getting 14 or fewer. And, pbinom(9, 23, 0.42)
# of getting 9 or fewer, so
pbinom( 14, 23, 0.42) - pbinom( 9, 23, 0.42)
# is our answer.
#
#  What is the probability of getting fewer than
#  6 or more than 14 successes out of 23 trials 
#  when the probability of success is 0.42?
pbinom( 5, 23, 0.42) + 
       pbinom( 14, 23, 0.42, lower.tail=FALSE)