Binomial Coefficient

Binomial coefficient: The formula used to calculate the probability for each possible outcome of a trial and to create the binomial probability distribution.

The binomial coefficient formula contains the variables N and r. The N represents sample size or the total number of trials. As an example, we will use a hypothetical study of jail inmates who have been booked on arrest and are awaiting word as to whether they will be released on bail or whether they will remain confined while they await criminal-court proceedings. The Bureau of Justice Statistics (BJS) reports that 62% of felony defendants are released from jail prior to the final disposition of their case (Reaves, 2013), so p = .62. Let’s say we draw a random sample of five recently arrested jail inmates. This means that for the binomial coefficient formula, N = 5.

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In the binomial coefficient, r represents the number of successes or, in other words, the number of times the outcome of interest happens over N trials. A researcher decides what “success” will mean in a given study. In the current example, we will define success as a defendant obtaining pretrial release. There are multiple values that r takes on in any given study, since there are multiple possible successes. If we wanted to find the probability that three of the five defendants would be released, we would input r = 3 into the binomial coefficient formula; if we wanted the probability that all five would be released, then r = 5.

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