Normal approximation to the binomial method of estimating a probability p

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A Binomial(n, p) has a mean and standard deviation given by:

From Central Limit Theorem, as n gets large the number of observed successes s will tend to:

From Central Limit Theorem Equation 2 for the binomial method can then be rewritten and p can be approximated by a Normal when n is large, as follows:

                              (1)

which can be rearranged to:

                              (2)

and which results in the following equation:

                          (3)

Figure 1: Example of Equation 3 estimate of p where s = 5, n = 10

Figure 2: Example of Equation 3 estimate of p where s = 1, n = 10

Equation 3 works nicely in the plot above for small n (10) because the number of successes was half of n, and so the uncertainty distribution is symmetric about 0.5, which nicely matches the properties of a Normal distribution. However, if one had observed just 1 success from 10 trials, it would look quite different, as shown in Figure 2: now the Normal approximation of Equation 3 is completely inaccurate, assigning considerable confidence to negative values, and fails to reflect the asymmetric nature of the uncertainty distribution.