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;
  3. you start within 3 attempts;
  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.

Example 1

P(X = 1) = 1/6

2. 'you need 4 attempts to start' requires you to fail for the first 3 attempts.

Example 2

3. 'starting within 3 goes', means you could start on your first, second or third goes

Example 3

4. 'you need greater than 6 attempts' is best thought of as being the same as needing 6 failures.

Example 4

The last part of this example gives us a special result to remember:

P(X > r) = qr

If:

X ~ Geo(p)

Then:

Expectation and variance

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.

Example

Hence:

σ = 1.94 (3 sf)

S-cool exclusive!!