List of basic formulas related to the binomial distribution:

  1. 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)!} ).
  1. Expected value (mathematical expectation):
    [
    E(X) = n \cdot p
    ]
  2. Variance:
    [
    Var(X) = n \cdot p \cdot (1 – p)
    ]
  3. Standard deviation:
    [
    \sigma = \sqrt{Var(X)} = \sqrt{n \cdot p \cdot (1 – p)}
    ]

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