The Geometric Distribution
The geometric distribution models a situation that involves having a series of attempts before the first success. When the player succeeds, the game stops because the outcome is decided – the object of the game maybe to succeed in the least number of attempts. You could be trying to score a six with a dice, get a hole in one or put a nail on a donkey at a children's party for example. Any situation to be modelled by the geometric distribution must satisfy the following two conditions.
Each attempt must be independent of the others. This implies that games of skill or learning are excluded, such as golf, the second example given above. Each attempt will not be independent because the participants will tend will get better as time passes.
The probability of success is constant. For instance, the probability of Manchester United scoring a goal in a game of football is not constant for every game. They will tend to score less goals when they play against better teams.
A typical sequence of failures followed by a success may be written as FFFFFFFFFFFFS – that is 12 failures followed by a success. If the probability of success is
then the probability of failure is
and so the probability of it taking thirteen attempts to succeed is
In general, for a geometric distribution, if n attempts are made with a player succeeding at the last, nth attempt and then stopping, there will be
failures followed by 1 success and the probability of someone succeeding at the nth attempt is
We write the Geometric distribution as
and
This is an important point: If the probability of success isthe expected number of throws before the first success is
Example: The probability of scoring a six on a fair dice is
a)What is the probability of throwing the dice 4 times before scoring a six?
b)What is the probability of having to throw the dice more than three times?
c)How many times would you expect to have to throw the dice before you scored a six?
a)Four failures followed by one success:
b)
c)Expected number of throws