Two jars, one secret
Two jars of candy, Jar A and Jar B — each with its own mix of orange and blueberry, no secret there. What's secret: a hidden coin flip picks which jar a candy comes from (heads = Jar A, tails = Jar B). Since you don't see the flip, your best guess starts at a plain 50/50 — that's your starting hunch.
Jar A
Jar B
Make the coin unfair — optional
A real coin lands heads about half the time, so most cases start as a plain 50/50 hunch. But you don't actually need a fair coin for any of this to work — drag this to make Jar A the coin's favorite (or underdog) before the case even starts.
Set up your jars, then start the case — the coin flips the moment you do.
What did you get?
Once the case starts, a candy gets pulled from whichever jar the coin picked — you only see its color, never which jar it came from. Every color you see is a clue. Draw a few and watch the case log fill up.
Why the hunch moved
Draw a candy to see this fill in. Here's the trick: imagine the case happened 100 times instead of once, and count how many of those 100 imaginary cases match what you actually saw.
Top rows = imaginary cases from Jar A, bottom rows = from Jar B. Each dot shows the candy that case would pull. Draw a candy above, and the dots matching what you saw get ringed — count where most of the rings land.
How this actually works — the formula, and its real name
Everything above is Bayes' theorem, just without the notation. Reverend Thomas Bayes worked out this exact idea in the 1740s — how to turn a starting guess into a smarter one once evidence shows up — and it's now behind things like spam filters, medical tests, and weather forecasts (see "Why this matters," below). If you ever see it written in a textbook, here's the dictionary between the words used above and the words a textbook uses:
| Plain word (used above) | Textbook name | Formal symbol |
|---|---|---|
| Starting hunch | Prior | P(A) |
| How likely your clue is, if it really is Jar A | Likelihood | P(clue | A) |
| How likely your clue is at all, either jar | Evidence | P(clue) |
| Updated hunch | Posterior | P(A | clue) |
Put together, that's Bayes' theorem itself:
P(A | clue) = P(clue | A) × P(A) ÷ P(clue)
The 100-dot picture and this formula compute the exact same number — the picture just lets you count it instead of multiplying fractions. That's also exactly what this page does under the hood every time you draw a candy: no rounding, no simulation, the real formula run on the real numbers from your jars.
Lock in your guess
Draw as many clues as you want, then guess which jar the coin actually picked. The hunch bar is showing you the odds — but the final call is yours.
Cases cracked: 0 / 0
You just did something doctors, weather forecasters, and spam filters do every day
Spam filters start with a hunch about whether an email is junk, then update it word by word — "free," "prize," and "click here" are all clues, the same way an orange candy was a clue about which jar it came from.
Weather forecasters start with a hunch based on the season and location, then update it as new radar and satellite readings — new clues — come in through the day.
Doctors start with a hunch based on how common a sickness is, then update it once a test result — a clue, and not always a perfectly certain one — comes back.
None of these are guessing games with candy jars, but every one of them is the exact same move you just made four times: start with a hunch, let a real clue change it, end up smarter than when you started.
The joke you just earned
Crack a case above and this fills in with the real swing your own hunch just made — a starting number, an ending number, and the punchline.
Crack a case above to unlock this.
A real number from a real update is worth putting on something — download it above, then upload it here to make it real.
Learn more
Thomas Bayes, An Essay towards Solving a Problem in the Doctrine of Chances (published 1763, two years after his death) — the origin of the theorem this page's name and math both come from. The 100-dot picture in step 3 is the "natural frequency" way of teaching it, popularized by psychologist Gerd Gigerenzer, specifically because it's proven to make this idea click for people who find the fraction-based formula confusing — count, don't calculate.