Binomial probability distribution

Definition

A binomial probability distribution represents the number of successes in a fixed number of trials, each with the same probability of success. It is used to model scenarios where there are only two possible outcomes.

5 Must Know Facts For Your Next Test

1. The binomial distribution is defined by two parameters: $n$ (the number of trials) and $p$ (the probability of success in each trial).
2. The mean (expected value) of a binomial distribution is calculated as $\mu = np$.
3. The variance of a binomial distribution is given by $\sigma^2 = np(1-p)$.
4. A scenario must meet three criteria to be modeled using a binomial distribution: a fixed number of trials, only two possible outcomes per trial, and independent trials.
5. The probability mass function for a binomial distribution is given by $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$, where $\binom{n}{k}$ is the binomial coefficient.

Review Questions

• What are the key parameters that define a binomial probability distribution?
• How do you calculate the mean and variance for a binomial distribution?
• What conditions must be met for an experiment to be considered a binomial experiment?

Related terms

Discrete Random Variable: A variable that can take on only specific, isolated numerical values.

Probability Mass Function (PMF): A function that gives the probability that a discrete random variable is exactly equal to some value.

Binomial Coefficient: $\binom{n}{k}$ refers to the number of ways to choose $k$ successes out of $n$ trials and is calculated as $\frac{n!}{k!(n-k)!}$.