# Stirling's formula probability

• indigojoker
This formula is commonly used to estimate large factorials. In summary, we can estimate the probability of all 50 states being represented in a group of 50 councilmen chosen at random using Stirling's formula, which gives us P=\sqrt{2 \pi 50} e^{-50}.
indigojoker
Use Stirling's formula, n!=sqrt(2 pi n) n^n e^{-n}, to estimate the probability that all 50 states are represneted in a group of 50 councilmen chosen at random.

I think it should be:

$$P=\frac{50!}{50^{50}}$$

So using Stirling's formula, we get:

$$P=\frac{\sqrt{2 \pi 50} 50^{50} e^{-50}}{50^{50}}$$
$$P=\sqrt{2 \pi 50} e^{-50}$$

is this the correct approach?

indigojoker said:
Use Stirling's formula, n!=sqrt(2 pi n) n^n e^{-n}, to estimate the probability that all 50 states are represneted in a group of 50 councilmen chosen at random.

I think it should be:

$$P=\frac{50!}{50^{50}}$$

So using Stirling's formula, we get:

$$P=\frac{\sqrt{2 \pi 50} 50^{50} e^{-50}}{50^{50}}$$
$$P=\sqrt{2 \pi 50} e^{-50}$$

is this the correct approach?
Presuming each councilman represents one of the 50 states, your work is correct!

## 1. What is Stirling's formula probability?

Stirling's formula probability is a mathematical formula that approximates the factorial of a large number. It is used to estimate the number of possible arrangements or combinations of a given set of objects.

## 2. How is Stirling's formula probability derived?

Stirling's formula probability is derived from the Stirling's approximation, which is a mathematical formula that approximates the factorial of a large number. It is based on the Gamma function and is commonly used in probability and statistics.

## 3. What is the purpose of using Stirling's formula probability?

Stirling's formula probability is used to simplify complex calculations involving large factorials. It provides a good estimation of the actual value and is often used in theoretical and practical applications in science and engineering.

## 4. Can Stirling's formula probability be used for all numbers?

No, Stirling's formula probability is only accurate for large numbers. As the number gets larger, the approximation becomes more accurate. For smaller numbers, it is better to use the actual factorial calculation.

## 5. Are there any limitations to using Stirling's formula probability?

Yes, there are limitations to using Stirling's formula probability. It is an approximation and may not always provide an accurate result. Additionally, it does not work well for numbers that are close to 0 or negative. It is best used for large positive numbers.

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