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# The Binomial distribution

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## The Binomial distribution

Suppose that an experiment consists of **n identical** and **independent** trials. For example flipping a coin over and over again **n** times. For each trial there are 2 outcomes.

'Success' - which is given probability **p**

'Failure' - which is given probability **q** where **q = 1 − p **

Then if X = the number of successes, we say that X has a binomial distribution.

**We write:**

This is sometimes written as: **X ~ B(n, p)**

If our random variable follows a binomial distribution, then the associated probabilities are calculated using the following formula:

**Note:** If you haven't seen

see the section on Combinations.

gives us the number of ways of choosing r objects from n and is calculated by:

You may also have a button on your calculator that will do all that for you.

**Let's see this in action...**

**Example:**

The probability that sixth formers know what the first prime number is, 0.35.

Find the probability that in a sample of 14 sixth formers, the number who know is...

- exactly 3;
- less than 3, and;
- greater than 1.

If we let X be the number of successes, our distribution is given by:

**Using the formula:**

Therefore:

Therefore:

To get P(X > 1), we **could** calculate this by working out:
P(X = 2) + P(X = 3) + P(X = 4)... + P(X = 14), taking a very long time and getting extremely bored! Instead, we use the fact that these distributions sum to 1 (they are **exhaustive!**)

Therefore:

**Question:**

See if you can work out the following probabilities if:

If **X ~ bin(n, p)**

Then:

- E(X) = n × p
- Var(X) = n × p × q

n = number of trials

p = probability of a success

q = probability of a failure = 1 − p

**Example:**

If X ~ bin(26, 0.2)

Then: E(X) = 26 × 0.2 = 5.2

Var(X) = 26 × 0.2 × 0.8 = 4.16