Question

How to find the probability when given the mean and standard deviation

Answer 1

You cannot do it unless you know the type of distribution. Usually, it is possible to write the exact formula of probability density function (PDF) using the mean and standard deviation. For example, if mean $\mu$ and standard deviation $\sigma$ is given, and the distribution is normal distribution, the PDF is

$\displaystyle f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{- \frac{(x-\mu)^2}{2 \sigma^2}}$

To find the probability that random variable x is less then certain value X, we need to calculate

$P(x \leq X) = \int_{-\infty}^X f(x)dx = F(X)$

This function $F(X)$ is called cumulative distribution function (CDF).

As an example, for normal distribution CDF is

$F(X) = \frac{1}{2} [1 + erf(\frac{x-\mu}{\sigma \sqrt{2}})]$

where erf() is called error function with tabulated values for different arguments (it can be calculated numerically in different softwares as well).

Also, $P(X_1 \leq x \leq X_2) = F(X_2) - F(X_1)$