Formula of Binomial Probability

A binomial experiment is one that possesses the following properties:

1. The number of successes X in n trials of a binomial experiment is called a binomial random variable.The experiment consists of n repeated trials;
2. Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial);
3. The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent.

The probability distribution of the random variable X is called a binomial distribution, and is given by the formula:

$\displaystyle{P}{\left({X}\right)}={{C}_{{x}}^{{n}}}{p}^{x}{q}^{{{n}-{x}}}$

where

n = the number of trials
x = 0, 1, 2, ... n
p = the probability of success in a single trial
q = the probability of failure in a single trial
(i.e. q = 1 − p)
$\displaystyle{{C}_{{x}}^{{n}}}$ is a combination

P(X) gives the probability of successes in n binomial trials.

Let's consider the experiment where we take a multiple-choice quiz of four questions with four choices each, and the topic is something we have absolutely no knowledge. Say... theoretical astrophysics. If we let X = the number of correct answer, then X is a binomial random variable because

• there are a fixed number of questions (4)
• the questions are independent, since we're just guessing
• each question has two outcomes - we're right or wrong
• the probability of being correct is constant, since we're guessing: 1/4

So how can we find probabilities? Let's look at a tree diagram of the situation:

Finding the probability distribution of X involves a couple key concepts. First, notice that there are multiple ways to get 1, 2, or 3 questions correct. In fact, we can use combinations to figure out how many ways there are! Since P(X=3) is the same regardless of which 3 we get correct, we can just multiply the probability of one line by 4, since there are 4 ways to get 3 correct.

Not only that, since the questions are independent, we can just multiply the probability of getting each one correct or incorrect, so P() = (3/4)³(1/4). Using that concept to find all the probabilities, we get the following distribution:

x	P(x)
0
1
2
3
4

We should notice a couple very important concepts. First, the number of possibilities for each value of X gets multiplied by the probability, and in general there are ₄C_x ways to get X correct. Second, the exponents on the probabilities represent the number correct or incorrect, so don't stress out about the formula we're about to show. It's essentially:

P(X) = (ways to get X successes)•(prob of success)^successes•(prob of failure)^failure