18.X.1. Definition

The Binomial Distribution is a discrete probability distribution that gives the probability of exactly k successes in n independent Bernoulli trials, each with a success probability p.

The Binomial Distribution specifically models scenarios where we conduct a fixed number of independent trials. Each trial can end in one of two outcomes: either success or failure. The parameter 'n' denotes the total number of trials being conducted, while 'k' represents the number of successes we are interested in. The parameter 'p' is the probability that any given trial results in a success.

Imagine you're rolling a die 10 times and you're interested in how many times you roll a '6'. In this scenario, the die rolls represent the independent Bernoulli trials. Getting a '6' is a success, and the probability of rolling a '6' on each individual trial is p = 1/6.

The probability mass function (PMF) is given by:

P(X = k) = (n choose k) * p^k * (1−p)^(n−k)

Where: • n: Number of trials • k: Number of successes (0 ≤ k ≤ n) • p: Probability of success in a single trial • 1−p = q: Probability of failure • (n choose k)= n! / (k!(n−k)!)

The PMF provides us with a mathematical formula to calculate the probability of achieving exactly 'k' successes in 'n' trials. The term (n choose k), often denoted as C(n, k), is a binomial coefficient that calculates the number of different ways we can achieve 'k' successes from 'n' trials. In the formula, p^k indicates the probability of success raised to the number of successes, and (1-p)^(n-k) represents the remaining trials which resulted in failure.

If we flip a coin 3 times and want to find the probability of getting exactly 1 head (success), the formula helps us determine how many different ways we can get 1 head out of 3 flips, while also calculating how these outcomes relate to their respective probabilities.

Where: • n: Number of trials • k: Number of successes (0 ≤ k ≤ n) • p: Probability of success in a single trial • 1−p = q: Probability of failure • (n choose k) = n! / (k!(n−k)!)

Each variable plays a crucial role in the PMF. 'n' indicates how many times an experiment is conducted. 'k' is bounded between 0 and 'n', meaning we can only have as many successes as the number of trials. The 'p' value changes based on the context of the experiment and significantly affects the probabilities calculated. 'q', the probability of failure, serves to provide a full view of outcomes since 'p' and 'q' together must equal 1.

Think of a basketball player who shoots 10 times in a game. Here, n = 10 (the total shots), p might be the player's free throw success rate (like 0.8), and k could be how many successful shots we want to predict from those 10 attempts.

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