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.

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.