The binomial distribution is a probability distribution that models the number of successes—and by extension, failures—in a fixed number of independent trials, each with the same probability of success.
The binomial distribution is applicable when an experiment meets the following criteria:
The binomial distribution is characterized by two parameters:
These parameters are crucial in determining the shape and properties of the distribution.
The probability of exactly $k$ successes in $n$ trials is given by the binomial probability mass function:
Consider an event that occurs $n$ times, with probability of success $p$ and probability of failure $1-p$. We want to find the probability of getting exactly $k$ successes. Let us call the event of success $S$ and the event of failure $F$.
Because there are multiple different orders that this can happen, for example:
$$SSSFF \cdots$$
$$SFSS \cdots$$
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