List of formulas related to independent trials and Bernoulli’s formula:

  1. Bernoulli’s 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. Independent trials:
    If the trials are independent, then the probability of the joint occurrence of events ( A ) and ( B ) is defined as:
    [
    P(A \cap B) = P(A) \cdot P(B)
    ]
  2. Total probability for independent trials:
    For ( n ) independent trials, the probability that an event will occur ( k ) times can be expressed using Bernoulli’s formula as above.

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