Stirling's Formula

Often when we calculate probabilities that use factorials of large numbers, we are only interested in the magnitude of the number or an approximation given in scientific notation. Compare the following:

Certainly we could round off the exact value of n!, but the calculation of n! is bothersome, especially if we only need an approximation. The problem is to find a method for approximating n! without actually performing the factorial calculation. Stirling's Formula is an easy to use approximation.

Stirling's Formula:

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In practice, we apply Stirling's formula by first writing it in log form,

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and then exponentiating the right-hand side.

For example: if n = 52.

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Compare this to the result given by the computer:

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How good is Stirling's Formula?

Although two numbers may have the same magnitude, they may still be quite far apart in actual value.

With computers and calculators so available, do we really gain anything from Stirling's Formula?

...After all, it is easier to type n! = than Stirling's Formula =.

Stirling's Formula has been is used to prove important results in probability theory