Bertrand’s Box Paradox Calculator
See how observing one coin from a randomly chosen box updates the probability that the other coin in the same box matches it, the classic Bertrand’s box paradox.
What this calculator does
Bertrand’s box paradox is a classic probability puzzle. Imagine boxes that each hold two coins: some boxes have two gold coins, some have two silver coins, and some have one of each. You pick a box at random, draw one coin, and see its color. This calculator finds the probability that the other coin in that same box matches the color you saw.
How the calculation works
The key is to count draws, not boxes. Each two-gold box produces two possible “gold coin drawn” events (either coin could be drawn first), while each mixed box produces only one “gold coin drawn” event. The probability the other coin also matches is the number of matching draws divided by the total number of draws of that color: P(match) = (2 x same-type boxes) / (2 x same-type boxes + mixed boxes).
The counterintuitive result with one box of each type is that after seeing gold, the other coin is gold with probability 2/3, not 1/2 — because a two-gold box gives twice as many chances to draw gold as a mixed box does. This tool is for educational and general learning purposes.