Binomial
Probability
"Exactly", "At Most", "At Least"
Problem used for demonstration: 
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) can also be used to quickly determine "exactly". (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 to produce a list of the exact probabilities.
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.

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