More Probability Tools
Classic problems and calculators, among the most searched probability topics worldwide.
Birthday Paradox
In a group of N people, what's the probability at least two share a birthday? The answer is shockingly high.
With only 23 people, the probability already exceeds 50%. With 70+ people it's essentially certain.
Lottery Odds Calculator
How rare is a lottery jackpot? Adjust the pool size and how many numbers you pick.
C(49,6) = 13,983,816 combinations.
Normal Distribution / Z-Score
Given a normally distributed population, what percentage fall above or below a given value?
Z = (130 − 100) / 15 = 2.000. Within normal range (±2σ).
Expected Value Calculator
Calculate the long-run average outcome of any probabilistic event by entering all possible outcomes.
Prob sum: 1.0000 ✓
Monty Hall Problem
You pick 1 of 3 doors. The host reveals a goat. Should you switch? Run the simulation to see why the answer surprises everyone.
Geometric Distribution
How many trials until your first success? Used in quality control, network retries, and sports streaks.
P(X = k) = (1−p)^(k−1) · p
Bayes' Theorem Calculator
If a test is positive, what's the real probability you have the condition? Bayes' theorem is the basis of all Bayesian reasoning.
(base rate, e.g. 0.01 = 1%)
(e.g. 0.95 = 95%)
(e.g. 0.95 = 95%)
With a base rate of 1.0%, a positive test result only means a 16.1% real chance. This illustrates why rare disease testing produces many false positives.
Confidence Interval Calculator
Given a sample mean, standard deviation, and sample size, compute the interval that contains the true population mean with a chosen probability.
Margin of error: ±5.368. With 30 samples, there is a 95% chance the true mean falls in this range.
Formula: x̄ ± z* · (σ / √n), where z* = 1.96 for 95% confidence.
Poisson Distribution
Models the probability of a given number of events occurring in a fixed interval when the average rate is known. Used in queuing, traffic, reliability engineering.
P(X ≤ 3) = 64.72%
Mean = λ = 3, Variance = λ = 3
P(X = k) = e^(-λ) · λ^k / k!
Central Limit Theorem Simulator
No matter the shape of the underlying distribution, the distribution of sample means approaches a normal distribution as sample size grows. See it live.
Draws 500 samples of size 30 and plots the distribution of their means.