Exploring Probability
Interactive simulations of games where the average outcome and the typical outcome disagree.
Ask what a gamble is worth and you will usually be handed one number: the average. It is a good number. It is also a summary of something much larger — a whole distribution of futures, most of which you will never see — and summaries leak. An average assumes the outcomes add up: that the good years and the bad years sit side by side and cancel. Real games rarely oblige. Outcomes multiply, so one bad round scales everything that comes after it. Losses absorb, so $0 is a place you can arrive at but never leave. Tails run off toward infinity, so the average ends up carried entirely by outcomes that will never actually happen to you.
When any of that is true, the average stops describing anybody. The typical player and the average player part company, and the gap between them is where the interesting mathematics lives — and where most bad financial decisions get made. Each game below is one place that gap opens up. They are simulated live in your browser, so you can push the numbers around and watch the gap open and close yourself.
Too good to be true
Heads pays +50%, tails costs 40% — that's +5% a round forever, in expectation. Play it yourself and you go broke almost surely. Both are true at once, and reconciling them is the whole subject.
Check out the ergodic coin flip →How much to bet
With the odds fixed, bet size alone decides whether you finish ahead or broke. Bet too little and you barely grow; too much and swings wipe you out. The right size is a formula, not a feeling.
Check out Kelly bet sizing →Nearly fair is not fair
With $100 against a house with millions, even a nearly fair coin ruins you almost certainly — and a perfectly fair one ruins you with certainty. Against a losing edge, betting bigger is safer than betting small.
Check out gambler's ruin →A ticket worth infinity
A coin is tossed until it lands tails, doubling the pot each time it doesn't. The expected payout is infinite, yet almost nobody would pay more than a few dollars — and they're right not to.
Check out the St Petersburg paradox →Winning by never winning
Betraying beats cooperating no matter what the other player does, so both betray and both lose. Play it hundreds of times against every strategy imaginable, and the winner is four lines long — and never once outscores an opponent.
Check out the iterated prisoner's dilemma →Should you switch?
Three doors, one prize; the host, who knows where it is, opens a goat and offers a switch. It looks like fifty-fifty. It isn't: switching wins two times out of three, and thousands of readers refused to believe it.
Check out Monty Hall →Money from a stock going nowhere
Split your money between a stock and cash. The stock ends the year exactly where it started, yet regularly rebalancing back to half and half leaves you ahead anyway — you've harvested its volatility, not its direction.
Check out Shannon's demon →A bad bet both sides are right to take
Insurance premiums always exceed the expected payout, so buyers who maximize expected value should never buy any. Yet a band of premiums exists where both buyer and seller come out ahead — expected value alone misses it.
Check out insurance and risk pooling →Two losing games, mixed into a winner
Two games, each individually guaranteed to lose. Alternate between them and your capital climbs. It's not a trick — Juan Parrondo built it from the same physics as Feynman's ratchet: "losing" isn't a property a game carries by itself.
Check out Parrondo's paradox →A 95% accurate test, and a 2% chance you're sick
A 95%-accurate test comes back positive. Harvard staff and students, asked the odds you're actually sick, mostly guessed 95%. The real answer is about 2% — the missing piece is how rare the disease was to begin with.
Check out base rates and the 95% test →Twenty-three people, better than even odds
How many people need to be in a room before two probably share a birthday? Intuition says 183; the real answer is 23, because every pair is compared against every other pair — 253 comparisons, not one.
Check out the birthday problem →Reject the first 37%, then take the next record
Candidates arrive one at a time; accept or reject on the spot, no going back. Skip the first 37% no matter how good they look, then take the next one better than all of them — nothing does better.
Check out the secretary problem →Whatever you find, swapping looks 25% better
One envelope holds twice what the other does. Find $100, and swapping looks worth $125 — but the same logic applies to any amount, even before you look, so you'd swap forever. The flaw hides in an unstated prior.
Check out the two-envelope paradox →Keep testing until it works
Test as you go, adding data until the result is significant. It feels diligent; it's actually a false-positive machine — a nominal 5% error rate can climb past 60% once you're allowed to keep looking.
Check out optional stopping →