# Binomial Probability of Success

Given a random sample of n items and a probability of failure/defect rate p, this tool calculates the probability that exactly x failures will occur in the sample.
The probability that exactly x failures will occur in a random sample of n items is given by:

The tool also calculates the cumulative probability that not more than r failures will occur in a sample of n items. The cumulative probability that r or fewer failures will occur in a sample of n items is given by:

where q = 1 - p.

For example, a manufacturing process creates defects at a rate of 2.5% (p=0.025).
A sample of 20 parts is randomly selected (n=20).
What is the probability that the sample contains 3 or fewer defective parts (r=3)? The probability of finding 3 or fewer defects in 20 samples is 0.9986.

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Calculation Inputs: