Bertrand's Box Paradox Calculator
Bertrand's box paradox is a classic probability puzzle that trips up almost everyone on first meeting. You have boxes, each holding two coins. Some boxes hold two gold coins, some hold a gold and a silver, and some hold two silver. You pick a box at random, reach in and draw one coin without looking at the other, and it turns out to be gold. What is the chance that the coin still in the box is also gold? The intuitive answer of one half is wrong. The correct answer for the classic setup is two thirds, because drawing a gold coin is stronger evidence that you reached into a two-gold box than into a mixed one: the two-gold box offers two chances to draw gold, the mixed box only one. This calculator lets you set how many boxes of each type are in play and then works the exact conditional probability by counting gold coins, the clean way to see past the illusion. It also shows the answer as a tidy fraction, the number of gold coins that could have been drawn, and the overall chance your first draw is gold. Change the mix of boxes to explore how the answer shifts, and use it alongside the how it works and worked example sections below to follow the Bayes reasoning step by step. It is a teaching tool for conditional probability, the same trap that appears in the Monty Hall problem and in medical test false positives.
The result assumes you pick a box uniformly at random and then draw one of its two coins at random. Set a box count to zero to remove that type. Teaching tool.
How it works
The trick is to count gold coins, not boxes. A two-gold box contributes two gold coins and a mixed box contributes one, so the total number of gold coins that could have been drawn is two times the two-gold boxes plus one times the mixed boxes. Of those gold coins, the ones whose partner is also gold are exactly the gold coins inside two-gold boxes, which is two times the two-gold box count. Dividing gives the chance the other coin is gold. The chance of drawing gold in the first place is the gold coin total divided by all coins, which is twice the number of boxes. This is Bayes' theorem in disguise, with each coin equally likely to be the one you drew.
Worked example
Use the classic setup: one box with two gold, one with a gold and a silver, one with two silver. The gold coins in play are two from the two-gold box plus one from the mixed box, which is three. Of those three gold coins, two live in the two-gold box where the partner is gold too. So the chance the other coin is gold is two out of three, which is 66.67 percent. The overall chance of drawing gold on the first pick is three gold coins out of six coins total, which is 50.00 percent. Notice the answer is two thirds, not the one half most people guess.
Related calculators
- Bertrand's Paradox Calculator: the random chord version.
- Bayes Theorem Calculator: update a probability with evidence.
- Conditional Probability Calculator: probability given a condition.
- Probability Calculator: single and combined events.
- Coin Flip Probability Calculator: heads and tails odds.