Binomial Cumulative Density Function

Last week I posted about the Binomial probability density function, and it is useful when calculating the probability of exactly x successes out of n trials given p probability of success for each trial.

Well, what happens if you want to know the probability of 2 or more successes for example?

We could use the Binomial PDF formula to determine the probability of exactly 2 successes, then 3, then 4, etc. up to 8 (the example last week had 8 trials). and adding those provides us a probability of 2 or more success out of 8 trials.

This is tedious. Yet, it does describe the nature of the cumulative density function for the binomial distribution. It is just the sum of individual pdf calculations over the range of interest.

\displaystyle P\left( x,n,p \right)=\sum\limits_{i=0}^{x}{\left( \begin{array}{l}n\\i\end{array} \right){{p}^{i}}{{\left( 1-p \right)}^{n-i}}}

Turns out to be just as tedious as first outlined. Yet, we can sometimes reduce the number of calculations by remembering the the sum of all the probabilities for each x always tallies to 1. Therefore in this case were we are interested in the probability of 2 or more successes, we can calculate the values for zero and one success, sum those and subtract from one.

The probability of exactly 0 successes is 0.1678, and for exactly 1 success is 0.3355. Thus,  we sum these two values and subtract from one, 1-(0.1678+0.3355) = 0.4967. Or, there is a little less then 50% probability of having 2 or more successes.


This plot is the CDF for p=0.2 and n=8. It shows the probability of less then or equal to a specific number of successes. The value of 1 success is just a little above 50%, and it is the tally of 0.1678 + 0.3355 = 0.5033.

The tally for 2 or fewer success include the probability for exactly two success, 0.2936, thus is 0.7969. This particular CDF approaches one quickly at about 4 successes.

This entry was posted in II. Probability and Statistics for Reliability and tagged by Fred Schenkelberg. Bookmark the permalink.

About Fred Schenkelberg

I am an experienced reliability engineering and management consultant with FMS Reliability, a consulting firm I founded in 2004. I left Hewlett Packard (HP)’s Reliability Team, where I helped create a culture of reliability across the organization, to assist other organizations. Given the scope of my work, I am considered an international authority on reliability engineering. My passion is working with teams to improve product reliability, customer satisfaction, and efficiencies in product development; and to reduce product risk and warranty costs. I have a Bachelor of Science in Physics from the United States Military Academy and a Master of Science in Statistics from Stanford University.

9 thoughts on “Binomial Cumulative Density Function

    • Hi Gary,

      Thanks for the note and congratulations on passing!

      Job well done.



      PS: be sure to follow this blog and join the Linkedin CRE Preparation group to help others preparing for the exam.

    • A simple real world issue is how many spares should I take with me to a site knowing that, say 20% of the spares are bad (and I can’t tell until it’s installed – which is a different problem to solve).

      p is 0.8 or the probability that the spare will work.

      If I only take one spare, there is a 20% chance it will fail. So, how many should I carry to be say, 95% or more certain to have a working spare with me. One way to sort this out is to calculate the binomial CDF and when the chance of at least one working out of the minimum number is at least 95% then I have the number of spares to take along.

      that is just one example, if anyone has others, please comment.



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