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# The Geometric distribution

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## 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) = q^{2} × p |
2 failures then success |

P(X = 4) = q^{3} × p |
3 failures then success |

P(X = r) = q^{r-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:**

- you start on your first go;
- you need 4 attempts before you start;
- you start within 3 attempts;
- 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) = q ^{r}**

**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)