# Binomial Distribution

A binomial distribution describes a population whose members take on only certain, discrete values. This is the case, for example, with the number of 13C atoms in cholesterol because a molecule of cholesterol can have two 13C atoms, but it can not have 2.5 atoms of 13C. A population is continuous if its members may take on any value.

The probability of finding an atom of 13C in a molecule of cholesterol is given by the binomial distribution equation

where P(X,N) is the probability that an event will occur X times during N trials, and p is the event’s probability in a single trial. Carbon has two stable, non-radioactive isotopes, 12C and 13C, with relative isotopic abundances of, respectively, 98.89% and 1.11%.

Cholesterol has a chemical formula of C27H44O. To determine the probability of finding two atoms of 13C in a single molecule of cholesterol, P(2,27), we take X as 2, N as 27, and p as 0.0111, obtaining a probability of 0.0033, or 0.33%. The figure below shows the binomial distribution through five atoms of 13C.