About Geometric Distribution Calculator
This geometric distribution calculator models the number of trials needed to get the first success. It counts the successful trial itself, so the smallest value is 1. It opens on p = 0.2 and finds the chance of succeeding within 5 trials. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The distribution is memoryless: after any run of failures, the number of extra trials needed has the same distribution as at the start. The average wait is 1 ÷ p trials, so p = 0.2 means 5 trials on average.
Some books count the failures before the first success instead, which shifts every value down by one. This calculator uses the trial-number version, with k = 1, 2, 3 and so on.
How to use Geometric Distribution Calculator
- Choose the distribution from the list. Each has its own parameters, such as the mean and standard deviation for the normal.
- Enter the parameters. The hint under each field says what is allowed.
- Choose what to calculate: a lower tail, an upper tail, an interval, a density or point probability, or the inverse.
- Enter x, or a and b for an interval, or the probability for the inverse. The result updates as you type.
- Read the answer, the mean and variance, and the shaded chart. Open the working to see which CDF values were used.
- Copy the result with the copy button.
Features
- Nine distributions: normal, binomial, Poisson, geometric, exponential, uniform, Student t, chi-square and F.
- Lower and upper tails, strict and non-strict, and intervals P(a ≤ X ≤ b).
- Density for continuous and point probability for discrete distributions.
- Inverse (quantile) calculation, including critical values for t, chi-square and F.
- Mean, variance and standard deviation, shown as undefined where they do not exist.
- Bell curve with shaded tail or interval for continuous distributions, and a bar chart for discrete ones.
- Upper tails computed directly, so very small p-values keep their precision.
- Formula of the distribution in KaTeX and a short worked step for every answer.
Geometric distribution
The first success on trial k means k - 1 failures followed by one success.
P(X = k) = (1 − p)^(k − 1) × p
P(X ≤ k) = 1 − (1 − p)^k
mean = 1 / p
variance = (1 − p) / p²Worked examples
p = 0.2: P(X ≤ 5) gives 0.67232
1 - 0.8^5 = 0.67232, the chance of at least one success in five tries.
p = 0.2: P(X = 3) gives 0.128
Two failures then a success: 0.8 × 0.8 × 0.2 = 0.128.
Privacy: how it works
- Everything is calculated by code running in your browser tab. Nothing you enter is sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- For a discrete distribution, P(X < 3) and P(X ≤ 3) are different. For a continuous one they are the same.
- For a two-sided critical value, such as the 95% value of t, put the inverse probability at 0.975.
- A density can be above 1 when the spread is small. It is not a probability.
- If your question is "at least k", use P(X ≥ x), not P(X > x).
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
Does the geometric distribution start at 0 or 1?
There are two conventions. This calculator counts the trial on which the first success occurs, so it starts at 1 and has mean 1 ÷ p.
What does memoryless mean?
Past failures do not change the future. The chance of needing more than 3 further trials is the same whether you have just started or have already failed 10 times.
How do I find the number of trials for a 90% chance of success?
Choose Inverse and enter 0.9. The result is the smallest number of trials with at least a 90% chance of one success.
More Probability Distribution Calculator pages
- Probability Distribution CalculatorPick a distribution, set its parameters and get PDF, CDF, interval probabilities, quantiles, mean and variance with a chart.
- Normal Distribution CalculatorEnter a mean and standard deviation to get tail probabilities, intervals and percentiles on a bell curve.
- Binomial Distribution CalculatorEnter the number of trials and the chance of success to get P(X = k), P(X ≤ k), P(X ≥ k) and ranges.
- Poisson Distribution CalculatorEnter the average rate λ to get the chance of exactly k events, at most k, at least k or a range.
- Exponential Distribution CalculatorEnter the rate λ to get the chance a waiting time exceeds, or stays below, a value, and its percentiles.
- Uniform Distribution CalculatorEnter the minimum a and maximum b to get the chance of landing in any sub-interval.
- Student t Distribution CalculatorEnter the degrees of freedom to get t critical values, tail areas and p-values on a shaded t curve.
- Chi-Square Distribution CalculatorEnter the degrees of freedom to get chi-square p-values, critical values and tail areas.
- F Distribution CalculatorEnter both degrees of freedom to get F p-values, critical values and tail areas.