Statistics/Distributions/Geometric

Geometric Distribution refers to the probability of the number of times needed to do something until getting a desired result. For example:

• How many times will I throw a coin until it lands on heads?
• How many children will I have until I get a girl?
• How many cards will I draw from a pack until I get a Joker?

Just like the Bernoulli Distribution, the Geometric distribution has one controling parameter: The probability of success in any independent test.

If a random variable X is distributed with a Geometric Distribution with a parameter p we write its probability mass function as:

$P(X=i)=p{(1-p)}^{i-1}$

With a Geometric Distribution it is also pretty easy to calculate the probability of a "more than n times" case. The probability of failing to achieve the wanted result is ${(1-p)}^k$.

Example: a student comes home from a party in the forest, in which interesting substances were consumed. The student is trying to find the key to his front door, out of a keychain with 10 different keys. What is the probability of the student succeeding in finding the right key in the 4th attempt?

$P(X=4)=\frac{1}{10}{(1-\frac{1}{10})}^{4-1}=\frac{1}{10}{\left(\frac{9}{10}\right)}^3=0.0729$