Binomial
Probability
"Exactly", "At Most", "At Least"
Problem used for demonstration:
A fair coin is tossed 100 times. What is the probability that: a. heads will appear exactly 52 times? b. there will be at most 52 heads? c. there will be at least 48 heads? 
We have seen that the formula used
with Bernoulli trials (binomial
probability) computes the
probability of obtaining exactly "r" events in "n" trials: We have also seen that the builtin
command binompdf (binomial probability
density function) (Remember, the function
binompdf is found under Here is our answer to part a. If you want more information at a
glance, this command can also be used
The formula needed for
answering part b is : There is a builtin command binomcdf (binomial cumulative density function) that can be used to quickly determine "at most". Because this is a "cumulative" function, it will find the sum of all of the probabilities up to, and including, the given value of 52. (The function
binomcdf is found under Here is our answer to part b. Again, if you want more information at a glance regarding cumulative probabilities, this command can be used to produce a list of the "at most" probabilities.
The formula needed for
answering part b is : Keep in mind that "at
least" 48 is the complement of "at most"
47. While there is no builtin command for "at least", you can quickly find the result by creating this complement situation by subtracting from 1. Just remember to adjust the value to 47. (Remember, the function binomcdf is found under
Again, if you want more information at a glance regarding cumulative probabilities, this subtraction process can be used to produce a list of the "at least" probabilities.
