Binomial coefficient

For a binomial variable #X#, the number of possible combinations of #x# successes in #n# trials equals \begin{equation*}{n\choose x}=\frac{n!}{x!(n-x)!}, \end{equation*}where #0\le x\le n# and #n!# (#n#-faculty) #=n(n-1)(n-2)\ldots 1#. By definition: #0!=1#.

Formula binomial distribution

For a binomial variable #X#, the probability of #x# successes in #n# trials is \begin{equation*} P(X=x)={n\choose x}p^x(1-p)^{n-x}, \end{equation*} where #p# is the probability of success.

Expected value and standard deviation

The expected value of a discrete random variabele #X# with #k# possible outcomes, equals the mean of the distribution \begin{equation*} \mu= \sum_{i=1}^k{x_iP(x_i)}. \end{equation*} The standard deviation of this random variable is \begin{equation*}\sigma= \sqrt{\sum_{i=1}^k{(x_i -\mu)^2P(x_i)}}.\end{equation*} For a dichotomous (binary) binomial variable simpler formulas are available. The expected value of probability of success #p# with #n# trials is \begin{equation*} \mu= np \end{equation*} and the standard deviation is \begin{equation*} \sigma= \sqrt{np(1-p)}. \end{equation*}