If you’re reading this page, you’re probably a math geek. Or maybe you’re just really into distributions and are already familiar with binomial, Poisson, negative binomial, hypergeometric, exponential, and normal distributions. If so: congrats! You’re basically one of those people who gets excited about things like going to the DMV! But seriously: these other distributions can all be useful at times, but today we’re going to focus on the geometric distribution. So what is it? How is it used? When is it used? If you’re looking for answers to these questions, you should keep reading this post.

What is the geometric probability distribution?

The geometric probability distribution is a discrete probability distribution that can be used to model the number of trials needed before the first success occurs. For example: when you flip a coin and want to know how many times you need to flip it before you get heads. Another example is: what is the number of times a machine is needed to produce a successful outcome, such as an ATM?

The geometric distribution calculator computes these parameters and plots them on a graph.

Mean of geometric distribution

The mean of geometric distribution is the probability of success or the number of trials needed for the first successful outcome. For example, if you roll a fair die 100 times, what is the probability that you will roll a 6 before rolling a 5? The mean number of trials required until you roll your first “6” is 25 (or 1/6).

To calculate this value, we need to use another formula:

\text {Mean} = P \times \frac {n}{n+1}

In this equation, P is the probability of getting a 6, and n is the number of completed trials.

Geometric distribution formula

The Geometric probability formula is:

\text {GP} = (1-\text {ps})^{nf} \times \text {ps}

In this equation, ps is the probability of success, and nf is the number of failures. If you’re rolling a fair die, with the goal of reaching a certain number, the probability is 1/6. The number of failures is the number of times you rolled the die before getting the number you desire. So, if you roll it 3 times, and on the third roll you get the number you want, the number of failures is 2.

How to use the geometric distribution calculator: an example

The geometric distribution is very easy to use because there are just two parameters you need to enter. Those parameters are the number of failures and the probability of success. From this, the calculator will give you the geometric probability, the mean, variance, and standard deviation.

The variance of geometric distribution

The variance of the geometric distribution is equal to:

V = \frac {(1 - \text {ps})} {ps^2}

For our previous example, rolling a fair die, the probability of success in this formula would be 1/6, so the variance would be equal to 30.

Geometric distribution examples

The geometric distribution is used to model the number of trials until the first success, and can also be used to model the number of failures until the first success.

We use it in situations where each trial (or failure) has an independent probability of succeeding. A coin toss would be an example of this type of situation because there is no correlation between how many times you’ve tossed a coin before, or if you’ve tossed it at all. The probability that a fair coin will land on heads after one toss is 50%. If we have 100 tosses with equal probability, then there will be a 50% chance that we won’t get any heads at all in those 100 tosses; however, if we get one head on our first try (i.e., in our first 100th trial), then there’s only a 25% chance that you will get heads on your next toss, then 12.5% on the next one and so on.

Geometric probability distribution

The geometric probability distribution is the probability of success on a single trial.

Let’s say you’re playing a game where you have to toss a coin and get it to land heads up three times in a row. The geometric probability distribution would be the chance that you will get tails on your first try, followed by two heads. In other words, if you were to flip the coin three times (or, in this case, land it on its side three times), the geometric probability distribution would tell you what your chances are of getting all heads or all tails in those three flips.

The standard deviation of geometric distribution

The standard deviation is a measure of the spread of data, which is the average of the square of the deviations from the mean.

We can use it to measure the variation of a data set. It describes how far each value in a data set is from its average. If a distribution has a large variance, there will be much more variation than if it has a small variance.

Binomial vs geometric distribution

The geometric distribution follows a different pattern than the binomial distribution. In fact, it’s a special case of the negative binomial distribution. The geometric distribution is also discrete and continuous, but it’s easier to understand with an example!

Suppose you have five marbles in your bag and they’re all blue. If you choose one marble at random and then replace it back into your bag without looking at its color, what are the chances that your next choice will be blue? To calculate this, you would use geometric distribution, using the number of marbles in the bag and the number of different colors of marbles.

The geometric distribution is a special case of the negative binomial

The geometric distribution is a special case of the negative binomial. The negative binomial can have a variable number of trials or a fixed number of trials. In contrast, all geometric distributions are fixed-trial distributions because they only have one trial.

For example, suppose that we want to know how many defective items will occur in each batch of 25 products manufactured by our factory. We could use a Poisson with 25 trials and 10 successes per batch, but this would require us to create separate sheets for different batches with different numbers of successes. By using Geometric instead, we can simply enter 1 defective item for every batch instead of creating multiple sheets and entering 10 defectives for each one!

FAQ

What is geometric distribution?

The geometric probability distribution is a discrete probability distribution that can be used to model the number of trials needed before the first success occurs.

Examples of geometric distribution

An example of geometric distribution is a coin toss because the probability of success is not influenced by how many times you have tossed the coin, or if you have tossed it at all.

What is the standard deviation?

The standard deviation is a measure of the spread of data, which is the average of the square of the deviations from the mean.