Probability Calculator
Free online probability calculator. 100% client-side, no upload.
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What is Probability Calculator?
A Probability Calculator computes the likelihood of events for common probability distributions. This tool covers four modes: the normal distribution (PDF, CDF, tail probabilities, z-scores, and interval probabilities), the binomial distribution (exact PMF and cumulative probabilities for n yes/no trials), the Poisson distribution (PMF and cumulative probabilities for a known rate lambda), and Bayes theorem (posterior probability from a prior, likelihood, and false positive rate). Everything is computed locally in your browser, so your inputs never leave your device.
Key Benefits
- Four distribution modes in one tool: normal, binomial, Poisson, and Bayes
- Normal mode: PDF, CDF, tail probabilities, z-score, and two-sided interval probability
- Binomial and Poisson modes: exact PMF, cumulative P(X <= k), P(X >= k), mean, and variance
- Bayes mode: posterior probability from prior, likelihood, and false positive rate
- 100% client-side - nothing leaves your device
Common Use Cases
- •Homework and exam problems in statistics and probability courses
- •Quality control: defect rates and binomial pass/fail analysis
- •Risk modeling and decision analysis with Bayesian updating
- •Estimating the probability of rare events with the Poisson distribution
How to Use the Probability Calculator
- Pick a distribution mode: Choose Normal, Binomial, Poisson, or Bayes from the mode selector. Each mode reveals its own set of inputs.
- Enter your parameters: For Normal, enter mean and standard deviation plus an x value (and an optional second value for interval probability). For Binomial enter n, p, and k. For Poisson enter lambda and k. For Bayes enter the prior, likelihood, and false positive rate.
- Read the results: The results panel shows the exact probability for your query. Normal gives PDF, CDF, and tail probabilities; Binomial and Poisson give the PMF and cumulative values; Bayes shows the posterior probability.
- Copy the output: Click Copy to copy every value for the active mode as plain text, ready to paste into a report or spreadsheet.
Key Features
- Four distribution modes: Normal, Binomial, Poisson, and Bayes theorem
- Normal: PDF, CDF (P(X < x)), P(X > x), z-score, and two-sided interval P(x1 < X < x2)
- Binomial: exact PMF, P(X <= k), P(X >= k), mean, variance, and std dev
- Poisson: exact PMF, cumulative probabilities, and rate parameters
- Bayes: posterior probability from prior, likelihood, and false positive rate
- 100% client-side - nothing is uploaded
Frequently Asked Questions
What is the difference between PDF, CDF, and PMF?
PDF (probability density function) gives the relative likelihood of a continuous random variable taking a specific value - it is not itself a probability. CDF (cumulative distribution function) gives P(X <= x), the probability of being less than or equal to a value. PMF (probability mass function) is the discrete equivalent of PDF and does return a probability for specific outcomes like k successes in a binomial trial.
When should I use the normal vs binomial vs Poisson distribution?
Use the normal distribution for continuous data like heights, test scores, or measurement errors. Use the binomial distribution when you count successes in a fixed number of yes/no trials, such as defect counts in a batch. Use the Poisson distribution when you count independent events over a fixed interval at a known average rate, such as calls per hour or typos per page. The normal can also approximate the binomial when n is large and p is near 0.5.
What is a z-score and how is it calculated?
A z-score measures how many standard deviations a value is from the mean, calculated as z = (x - mean) / stdDev. A z-score of 1 means the value is one standard deviation above the mean, which corresponds to roughly the 84th percentile in a standard normal distribution. Z-scores let you compare values from different normal distributions on a common scale.
How does Bayes' theorem work in this calculator?
Bayes' theorem updates a prior belief after seeing evidence. You enter the prior P(A) (your initial belief), the likelihood P(B|A) (probability of the evidence if A is true), and the false positive rate P(B|not A). The calculator computes the marginal probability P(B) and the posterior P(A|B) - the updated probability that A is true given the evidence. A classic use is medical testing, where a low prior and non-zero false positive rate often produce a surprisingly low posterior.