The de Moivre-Laplace Theorem in Probability Theory

The de Moivre-Laplace theorem (first published in 1738) is one of the earliest attempts to approximate probabilities by a normal distribution. This theorem provides a remarkably precise approximation of the distribution function (i.e., the cumulative probabilities) of a binomial distribution with parameters and . Those parameters correspond, for example, to the number of heads obtained in a sequence of tosses of a coin, where is the probability of a head in a single toss. Given the scarcity of calculation resources available in the century, this theorem proved at the time to be a very valuable tool.

For any choice of and , you will see the exact and approximate values of the probability of obtaining a number of heads between and (inclusive). Move the sliders to vary and as well as and . The orange curve is the cumulative normal distribution, while the staircase-shaped polygonal curve is the cumulative binomial distribution (a step function). The number of heads used in the calculation ( and ) appear in brown and blue on the top scale. The theorem is actually stated in terms of standardized values, that is, after subtracting the mean from each result and dividing by the standard deviation. The standardized values corresponding to the choices of and appear on the bottom scale.

As grows, you can see that the binomial approaches the normal, thus illustrating a special case of the central limit theorem of probability theory.