Bayes' Theorem with Dice

Update beliefs about dice based on evidence

Which Die Did I Use?

Die A

Fair (1/6 each)

Die B

P(6) = 1/3

You have Die A (fair, P(6) = 1/6) and Die B (biased, P(6) = 1/3). You pick one at random and roll a 6. What is the probability you picked Die A?

(picked at random)

(picked at random)

Given that you rolled a 6, the probability you were using Die A (the fair die) is 1/3 or approximately 33.3%.

P(Die A | rolled 6)

33.3%

P(Die B | rolled 6)

66.7%

We can verify this by computing P(B|6) directly: