List of basic formulas related to the binomial distribution:
- Probability formula:
[
P(X = k) = \binom{n}{k} p^k (1-p)^{nk}
]
where:
- ( P(X = k) ) is the probability that the event will occur exactly ( k ) times,
- ( n ) — total number of tests,
- ( k ) — the number of successful outcomes,
- ( p ) is the probability of success in one trial,
- ( \binom{n}{k} ) is the binomial coefficient, equal to ( \frac{n!}{k!(nk)!} ).
- Expected value (mathematical expectation):
[
E(X) = n \cdot p
] - Variance:
[
Var(X) = n \cdot p \cdot (1 – p)
] - Standard deviation:
[
\sigma = \sqrt{Var(X)} = \sqrt{n \cdot p \cdot (1 – p)}
]