Geometric Distribution Formula

Geometric distribution formula

The geometric distribution is a type of probability distribution, that deals with the number of successive failures before success is achieved in a Bernoulli trial. 

The Bernoulli trial is an experiment with only two possible outcomes: success or failure. It can simply be stated that in geometric distribution a Bernoulli trial is repeated until success is achieved and is only stopped after that. 

In geometric distribution, 

• The conducted trials are independent

• There can only be two results of the trial, success or failure

• The probability of success is denoted by p, and it is the same for each trial

Geometric distribution- Let’s assume a die is repeatedly rolled until number 4 is obtained. In this case the probability of getting 4, is 1/6. Here, the random variable X is used. This variable can take on several values, such as 1, 2, 3 until the first success i.e getting number 4 is achieved. So, here p = 1/6.

In case of geometric distribution, 

P (X = x) = (1 – p)^(x-1) p

P (X is less than equals to x) = 1 – (1 – p)^x. 

This function is also described by probability mass function (PMF) and the cumulative distributive function (CDF). Here, the probability of success is denoted by p and probability of failure is denoted by q. 

q = (1 – p) 

Solved examples

1) A patient is waiting for a suitable blood donor and the probability of getting a suitable match is 0.2. Find out the estimated number of donors who will be tested until a suitable match is found? 

Answer. In this problem, we are only looking for the success in geometric distribution. 

Now, the probability of success is 0.2.

So, the p = 0.2

E [X] = 1/p

= 1 / 0.2

= 5

So, the expected number of donors who will be tested until a suitable match is found is 5.

2) Let’s assume a boy is playing a game of darts. The probability of getting success is 0.4. Then what is the probability of hitting the bullseye on his third try? 

Answer. In this problem, we are looking for success, so here geometric distribution has to be used. 

p = 0.4

P (X = x) = (1 – p) ^(x-1)p

P (X = 3) = (1 – 0.4) ^(3-1) * 0.4

P (X = 3) = (0.6)² * 0.4

= 0.144

So, the probability that the boy will hit the bullseye on his third try is 0.144.