 # The Geometric distribution

## The Geometric distribution

If we let X be the random variable of the number of trials up to and including the first success, then X has a Geometric Distribution.

For example:

If you were to flip a coin wanting to get a head, you would keep flipping until you obtained that head. Or if you needed a double top in darts, you would keep throwing until you hit it.

The probabilities are worked out like this...

Remember:

p = probability of success

q = probability of failure

 P(X = 1) = p No failure,success on first attempt P(X = 2) = q × p 1 failure then success P(X = 3) = q2 × p 2 failures then success P(X = 4) = q3 × p 3 failures then success P(X = r) = qr-1 × p r - 1 failures followed by success

If X follows a Geometric distribution, we write:

X ~ Geo(p)

This reads as 'X has a geometric distribution with probability of success, p'.

Example:

In a particular game you may only begin if you roll a double to start. Find the probability that:

1. you start on your first go;
2. you need 4 attempts before you start;
4. you need greater than 6 attempts before starting.

Before we start, let's write down the distribution of X with the probability of a success - in this case - being the probability of rolling a double with 2 dice.

X ~ Geo (1/6) as probability of double = 1/6

1. 'starting on your first go' requires rolling a double on your first attempt. P(X = 1) = 1/6

2. 'you need 4 attempts to start' requires you to fail for the first 3 attempts. 3. 'starting within 3 goes', means you could start on your first, second or third goes 4. 'you need greater than 6 attempts' is best thought of as being the same as needing 6 failures. The last part of this example gives us a special result to remember:

P(X > r) = qr

#### Expectation and variance

If:

X ~ Geo(p)

Then: Example:

If the probability to pot a ball off the break in pool is 0.4, find the expected number of breaks before success and the corresponding variance and standard deviation.

Here:

X ~ Geo(0.4)

Therefore:

E(X) = 1/0.4 = 2.5

So we would expect to pot off the break every 2.5 goes. Hence:

σ = 1.94 (3 sf)