6.1 The Binomial Distribution
Consider a game where a card is randomly drawn from a deck of 52 playing cards 10 times, and each time you get to guess the suit of the card (each time, the randomly chosen card is returned to the deck). We can denote the the possible number of correct guesses to be \(X\) which can take any of the following values: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
We can use the Binomial Distribution to see your chances for each one of these values of \(X\):
We now introduce the Binomial Distribution formally: Suppose we have \(n\) "trials", each with an outcome of either "success" or "failure". Further suppose that for each trial, the probability of "success" is equal to \(p\), and that \(X\) is the number of "successes" from the \(n\) trials. We can model \(X\) using the Binomial Distribution. In mathematical notation, we define the distribution as \[ X \sim BIN(n, p),\] where:
- \(X\) is the number of successes
- \(n\) is the number of trials
- \(p\) is the probability of success for each trial.
If, for example, we are interested in the probability of some number \(x\) successes (out of the \(n\) trials), we can write this probability down as \(P(X = x)\).
Given in our playing cards example we have \(n = 10\) trials with a probability of "success" of \(p = 0.25\) for each trial, we define the distribution in this example as: \[X \sim BIN(10, 0.25).\]
Using this distribution, we can calculate the probability of particular events, for example:
- \(P(X = 0) = 0.056\)
- \(P(X = 1) = 0.188\)
- etc.
We will be learning how to calculate probabilities like these in lectures and computer labs.