Bayes' Theorem with Dice

Update beliefs about dice based on evidence

Calculator Dice Coins Marbles Cards Learn

Loaded Die DetectionLoaded Fair Given ResultFair?Which Die?Which?

P(Fair Die | Rolled Even)

✓

The Scenario

You have two dice: one fair, one biased toward even numbers (70% even). You pick one at random, roll it, and get an even number. What is the probability you picked the fair die?

Problem Setup

Fair Die

Picked with probability 0.5

Biased Die

Picked with probability 0.5

Step-by-Step Solution

Identify the Probabilities

(prior)

Calculate P(even) - Total Probability

Apply Bayes' Theorem

Answer

Given you rolled an even number, there is approximately a 41.7% chance you were using the fair die.

The evidence (rolling even) made the fair die slightly less likely, since the biased die is more likely to produce an even number.

Building Intuition

Imagine doing this experiment 100 times:

Pick fair die (50 times), roll even:50 × 0.5 = 25 times

Pick biased die (50 times), roll even:50 × 0.7 = 35 times

P(fair | even) =25/60 = 41.7%