## PROBABILITIES Course

# Binomial distribution

The binomial distribution $B(n,p)$ represents the probability of success on $n$ trials with a probability of $p$.

- The mass function is

\[
\begin{split}\begin{cases}
\mathbb{P}(X=k) = 0 & if & k > n \\
\mathbb{P}(X=k) = C_n^k * p^k * (1-p)^{n-k} & else \\
\end{cases}\end{split}
\]

- $\mathbb{E}(X) = \ n * p$
- $\mathbb{V}(X) = \ n * p * (1-p)$

The probability of having $k$ successes on $n$ trials means that

- we got $k$ successes
- we got $n-k$ failures (the remaining trials)
So we have the probability

- $p^k$ because we want $k$ successes with p the probability of success
- $(1-p)^{(n-k)}$ because we want $n-k$ failures and $1-p$ if the probability of failure.
- and since we don't care about the order, we need to multiply by the permutations $C_n^k$