A binomial experiment possesses the following properties: 1. The experiment consists of a fixed number, n, of identical trials. 2. Each trial results in one of two outcomes: success, S, or failure, F. 3. The probability of success on a single trial is equal to some value p and remains the same from trial
to trial. The probability of a failure is equal to q = (1 − p). 4. The trials are independent. 5. The random variable of interest is Y , the number of successes observed during the n trials.
Definition:
A random variable Y is said to have a binomial distribution based on n trials with success probability p if
and only if
$$p(y) = left( {begin{array}{*{20}{c}} n\ y end{array}} right){p^y}{q^{n - y}},;;y = 0,;1,;2,;...,;n;{rm{and}};0 le p le 1.$$
You can use the following Mathematica command to obtain the probability
Expection and Variance: Let Y be a binomial random variable based on n trials and success probability p. Then $$E(Y) = np\;\;\;{\rm{and}}\;\;\;V(Y) = np(1 - p)$$
You can use the following Mathematica command to obtain these results Expectation[x, x [Distributed] BinomialDistribution[n, p]] Variance[BinomialDistribution[n, p]]