Binomial experiments have

• a fixed number of trials \(n\), where
• each trial has only two possible outcomes, “success” (having probability \(p\)) and “failure” (having probability \(1 - p\)), where
  • “success” is defined as the condition which we are trying to satisfy, e.g. making a free throw, answering a question correctly, flipping a coin and getting “heads,”
• the “success” probability \(p\) is constant for all trials, and finally
• all trials are independent of each other.
  • A rule of thumb is we can assume independence if the sample size is less than or equal to 1/10th of the population size.

The binomial random variable \(X\) represents the number of “successes” in those \(n\) trials given probability of success \(p\). We can say “\(X\) is binomial with \(n = ...\) and \(p = ...\)” The trials in binomial experiments are known as Bernoulli Trials.

Probability Mass Function for a Binomial Random Variable

In a binomial experiment, we can count the number of possible outcomes having \(k\) successes: \begin{equation} \frac{n!}{k!(n - k)!} \tag{Number of combinations} \end{equation} which has a shorthand version: \begin{equation} \binom{n}{k} \tag{Number of combinations shorthand} \end{equation}

The pmf is a function that gives the probability that a discrete random variable is exactly equal to some value; thus we can express the probability distribution as \begin{equation} P(X = x) = \frac{n!}{x!(n - x)!} p^x (1 - p)^{n - x} , \ \text{for 0} \leq x \leq n \end{equation} where we’re multiplying the number of ways we can get \(x\) successes by the probability of getting \(x\) successes in \(n\) independent trials.

SciPy (a Python package) has a function to calculate the probability mass function of a random experiment:

  from scipy.stats import binom
  # probability of getting 2 heads out of 3 fair coin tosses
  binom.pmf(n=3, k=2, p=0.5)

Cumulative Distribution Function

This function answers the question “what’s the probability of getting at most \(k\) successes in a random experiment of \(n\) trials given success probability \(p\)?”

  # probability of getting at most 2 heads out of 3 fair coin tosses
  binom.cdf(n=3, k=2, p=0.5)

Mean and Variance

For binomial distributions, we have special-case formulas for the mean, variance, and standard distribution: \begin{equation} \mu_X = np \tag{Mean} \end{equation} \begin{equation} \sigma^2_X = np(1 - p) \tag{Variance} \end{equation} \begin{equation} \sigma_X = \sqrt{np(1 - p)} \tag{Standard Deviation} \end{equation}

Shape of Binomial Distribution

With a small number of trials \(n\) a \(p\) closer to 0 or 1 than to 0.5 will yield a skewed-right or skewed-left distribution, respectively; however as \(n\) increases, the distribution centers around \(p \cdot n\).