We repeat a Bernoulli experiment \(n\) times. Each time, we observe, if an event \(A\) occurred or not. Let the probability of \(A\) to occur be constantly \(p:=p(A)\) during each repetition of the experiment. Let \(X\) be the random variable counting the number \(k\) of the realizations of \(A\). Clearly, \(A\) can occur for \(k\) times with \(0\le k\le n\). The probability mass function of \(A\) occurring exactly \(k\) times is given by
\[p(X = k)=\begin{cases} \binom nk p^k(1-p)^{n-k}&\text{for }k=0,1,\ldots n\\\\ 0&\text{else.}\end{cases}\]
The binomial distribution (i.e. the probability distribution of the random variable \(X\)) is given by
$$\begin{array}{rcll} p(X \le x)&=&0&\text{for }x < 0\\ p(X \le x)&=&\sum_{k=0}^{k=x}\binom nk p^k(1-p)^{n-k}&\text{for }0\le x < n\\ p(X \le x)&=&1&\text{for }x \ge n\\ \end{array}$$
In the following interactive you can change the probability \(p\) of observing an event \(A\) in a Bernoulli experiment repeated \(20\) times. Changing this probability will change the probability mass function for different values of \(X\) (red) of observing \(A\) from \(0\) to \(20\) times and the probability distribution (blue):
Proofs: 1
Proofs: 1