Bayes' Theorem with Cards

Update beliefs about decks based on draws

Calculator Dice Coins Marbles Cards Learn

Which Deck?Which?Rigged DeckRigged?Deck TestingTest

Testing Deck Completeness

The Scenario

You find an old deck of cards. You want to test if it is complete (52 cards) or missing some cards. You draw cards and check if they are from certain categories.

The Hypotheses

Complete Deck

All 52 cards present

(prior)

Missing Face Cards

Only 4 face cards remain (8 missing)

(prior)

Test 1: You Draw a Face Card

Calculate P(face)

Apply Bayes' Theorem

Drawing a face card increases our confidence that the deck is complete from 70% to 85.7%.

Test 2: You Draw a Number Card (2-10)

Number Card Probabilities

Complete:

Missing:

Calculate P(number)

Apply Bayes' Theorem

Drawing a number card slightly decreases our confidence from 70% to 66.4%, because number cards are relatively more common in the incomplete deck.

Summary: Information Value of Cards

Card Drawn	Prior	Posterior	Effect
Face Card (J, Q, K)	70%	85.7%	+15.7% (evidence for complete)
Ace	70%	70%	No change (equal in both)
Number Card (2-10)	70%	66.4%	-3.6% (slight evidence against)

Key Insight

Different cards provide different amounts of information. Face cards are strong evidence for a complete deck, while number cards provide weak evidence against. The most informative draws are those that differ most between the two hypotheses.