Bayes' Theorem with Cards

Update beliefs about decks based on draws

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Is the Deck Rigged?

The Scenario

A magician claims to have a normal deck. You draw 3 cards with replacement and get 3 Aces in a row. What is the probability the deck is rigged (contains extra Aces)?

Assumptions

Prior: 5% of magicians use rigged decks
Rigged deck: 20% of cards are Aces (instead of 7.7%)
Cards drawn with replacement (shuffled back in)

Probability Setup

Normal Deck

Rigged Deck

Step-by-Step Solution

Calculate P(3 Aces) - Total Probability

Apply Bayes' Theorem

Answer

After seeing 3 Aces in a row, the probability the deck is rigged increases from 5% to approximately 48%.

Prior (before draws)

Posterior (after 3 Aces)

48%

Key Takeaways

1.3 Aces is ~17.5x more likely with a rigged deck (0.8% vs 0.046%)
2.This strong evidence nearly overcomes the low prior of 5%
3.We are still not sure (48%) - more draws would help