Bayes' Theorem with Cards

Update beliefs about decks based on draws

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Which Deck?Which?Rigged DeckRigged?Deck TestingTest

Which Deck Did I Draw From?

Standard Deck

4 Aces / 52 cards

Modified Deck

8 Aces / 48 cards

The Scenario

You have two decks: a standard 52-card deck (4 Aces) and a modified deck with 8 Aces (but only 48 cards total - no 2s). You pick a deck at random and draw an Ace. What is the probability you picked the standard deck?

Problem Setup

Standard Deck

Modified Deck

Step-by-Step Solution

Calculate P(Ace) - Total Probability

Apply Bayes' Theorem

Answer

Given you drew an Ace, there is only a 31.6% chance you used the standard deck.

P(Standard | Ace)

31.6%

P(Modified | Ace)

68.4%

Key Insight

The modified deck has twice as many Aces (8 vs 4), making it about twice as likely to produce an Ace draw. This shifts our belief strongly toward the modified deck when we observe an Ace.

Likelihood ratio: