As described in the referenced START sheet (Volume 7, Number 4), the binomial distribution is often applied to one shot devices. When applying the binomial, “p” is the proportion of the population that is defective. This parameter can be represented as the “unreliability”, or 1-R. In most cases the true value of “p” is unknown. Therefore, an estimator for “p” is calculated from a sample of the population using the expression r/n, where “r” is the number of defective items, and “n” is the sample size. As explained in the referenced START sheet, a confidence interval can be constructed on “p” using the F distribution if “n” and “r” are known (equations 4 and 5 for the lower and upper confidence bounds, respectively). For the lower bound, the number of denominator degrees of freedom (v2) is equal to 2r. If r=0 (no failures), then v2=0, and the confidence interval approximation using this method would be undefined. If a confidence interval on “p” is desired, several options can be considered. First, the sample size could be increased until at least one failure is observed. Second, one failure in the current sample can be assumed (conservative approach). Third, MIL-HDBK-189A discusses Monte Carlo techniques that can be applied. Finally, if “p” can be assumed, then the probability of observing 0 failures in a sample size of “n” can be calculated. The resulting probability is the confidence that the true proportion defective is less than or equal to the assumed “p”. This method is discussed in the START sheet entitled: “Operating Characteristic (OC) Functions and Acceptance Sampling Plans” (Volume 12, Number 1).

Richard Wisniewski,
Senior Reliability Engineer
Defense Systems Information Analysis Center (DSIAC)

Please note that the RIAC has been transitioned to the newly-formed Defense Systems Information Analysis Center (DSIAC). Inquiries related to the technical areas of reliability, maintainability and quality (RMQ) can continue to be made to the toll-free number at (877) 363-7422.