# 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)