List of formulas related to independent trials and Bernoulli’s formula:
- 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)!} ).
- 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)
] - 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.