Law of large numbers · normal distribution · expected value · Bayes · client-side

probability visualizerstatistics visualizernormal distribution calculatorexpected value calculatorBayes theorem calculatorlaw of large numberscoin flip simulator

//// Educational · Statistics

Probability & Statistics Visualizer

Coin flips, bell curves, lottery math, Bayes' theorem — the core ideas of statistics made visual and interactive.

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🪙 Coin Flip Simulator — Law of Large Numbers

Flip a coin 10 times — you might get 7 heads. Flip 1,000 times — you'll be very close to 50%. This is the "Law of Large Numbers." More trials = more predictable outcome.

Hit the button to run all 5 simulation sizes at once.

🔔 Normal Distribution — The Bell Curve

Tons of things in nature follow a bell curve: heights, test scores, measurement errors. The curve tells you how likely any value is. About 68% of values fall within 1 standard deviation (σ) of the mean — and 99.7% within 3σ!

Mean (μ)50

Std dev (σ)10

±1σ

68.27%

40 – 60

±2σ

95.45%

30 – 70

±3σ

99.73%

20 – 80

🎰 Expected Value — Should You Play the Lottery?

Expected value (EV) tells you the average outcome if you played many, many times. Almost all lotteries have negative EV — you lose money on average. See it yourself!

Ticket cost ($)

Prize ($)

Odds (1 in X)

Expected value per ticket

$-1.5000

You lose $1.5000 on average per ticket. Over 100 tickets: −$150.00

Win probability

1 / 10,000,000

= 0.000010%

Expected winnings

$5,000,000 × (1 / 10,000,000)

= $0.5000

Expected loss

$2 × (1 − 1/10,000,000)

= −$2.0000

Expected value (EV)

$0.5000 − $2.0000

= $-1.5000 per ticket

#ShowYourWork

🧬 Bayes' Theorem — The Medical Test Paradox

A test is 99% accurate. You test positive. What's the chance you actually have the disease? Probably much less than 99%! This is Bayes' theorem — it factors in how rare the disease is. A rare disease + lots of people tested = lots of false positives.

Disease prevalence (%)0.1%

Test sensitivity — true positive rate (%)99%

Test specificity — true negative rate (%)99%

Prior (prevalence)

0.1%

P(disease)

True positive

0.099%

P(+ and disease)

False positive

0.999%

P(+ and no disease)

If test positive…

9.0%

actual disease chance

Key insight: Even with a positive test, there's only a 9.0% chance you have the disease. Most positives are false alarms when the disease is rare.

Prior P(disease)

0.1%

= 0.1000%

True positive P(+ | disease)

sensitivity × prior = 99% × 0.1%

= 0.0990%

False positive P(+ | no disease)

(1 − specificity) × (1 − prior) = 1% × 99.9%

= 0.9990%

Total P(positive test)

true positive + false positive

= 1.0980%

Posterior P(disease | positive test)

true positive / total positive

= 9.02%

#ShowYourWork