Probability Basics: From Coin Flips to Real-World Decisions

This used to be a back-of-the-envelope job. Now it is one tap, but the logic has not changed. Probability isn't intuition's strong suit. The Monty Hall problem alone has fooled mathematicians.

The scale

0 = impossible, 1 = certain. Everything else is in between.

Independent events

Multiply: P(two heads in a row) = 0.5 × 0.5 = 0.25.

Conditional probability

P(A|B) = P(A and B) / P(B). The basis for Bayes' theorem.

Pro tips

• Gambler's fallacy: past coin flips don't change the next one. Each is independent.
• Base rates matter — 99% accurate medical test for a 1-in-10,000 disease still has more false positives than true positives.

Frequently asked questions

What's expected value? Sum of (outcome × probability). The long-run average if you repeated forever.

Why is Bayes' theorem so important? Updates probability as new evidence arrives — basis for modern ML and medical reasoning.

Whatever number you land on, sanity-check it against a second method or a quick estimate in your head.