Probability & Statistics Tools

Explore classic probability and statistics problems with interactive calculators and simulations. Test the Birthday Paradox, lottery odds, Bayes’ theorem, expected value, confidence intervals, Poisson distributions, the Central Limit Theorem, and more, all free in your browser.

Birthday Paradox

In a group of N people, what's the probability at least two share a birthday? The answer is shockingly high.

People in the group23

2366

With only 23 people, the probability already exceeds 50%. With 70+ people it's essentially certain.

50.7%

Probability of a shared birthday

0%50%100%

Lottery Odds Calculator

How rare is a lottery jackpot? Adjust the pool size and how many numbers you pick.

Pool size49

1090

Numbers to match6

110

Odds: 1 in13,983,816

Probability7.151e-6%

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?

Mean (μ)

Std Dev (σ)

Value (X)

Z-Score2.000

Below X97.72%

Above X2.28%

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.

Probability (0–1)

Value / Payout

Expected Value+0.000

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: probability of success per trial0.30

Mean trials3.33

P(1st try)30.0%

P(≤5 tries)83.2%

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.

Prior probability

(base rate, e.g. 0.01 = 1%)

Sensitivity (true positive rate)

(e.g. 0.95 = 95%)

Specificity (true negative rate)

(e.g. 0.95 = 95%)

Posterior Probability: P(condition | positive test)

16.1%

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.

Sample Mean (x̄)

Std Dev (σ)

Sample Size (n)

Confidence Level

95% Confidence Interval

[94.632, 105.368]

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.

λ (average events per interval)3

0.515

P(X = k) for k =3

22.404%

P(X = 3)

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.

Population Distribution

Sample size n30

1100

Draws 500 samples of size 30 and plots the distribution of their means.