The binomial distribution has its applications in experiments in probability subject to certain constraints. These are:
- There is a fixed number of trials – for example toss a coin 20 times.
- The outcomes are independent and there are just two possible outcomes-in the example I will use, these are head and tail.
- The probability of a head plus the probability of a tail must equal 1.
- The probability of 8 heads and 12 tails would be 20C8 x P(H)^8 x P(T)^12.
- Now any experiment in which the outcomes are of just two kinds and whose probability combined equals 1, can be regarded as binomial.
Data from the analyses of reference samples often must be used to determine the quality of the data being produced by laboratories that routinely make chemical analyses of environmental samples. When a laboratory analyzes many reference samples, binomial distributions can be used in evaluating laboratory performance. The number of standard deviations (that is, the difference between the reported value and most probable value divided by the theoretical standard deviation) is calculated for each analysis. Individual values exceeding two standard deviations are considered unacceptable, and a binomial distribution is used to determine if overall performance is satisfactory or unsatisfactory. Similarly, analytical bias is examined by applying a binomial distribution to the number of positive and negative standard deviations.