Not Playing With a Full Deck?

A standard deck of 52 playing cards has 4 suits: clubs (♣), spades (♠), hearts (♥) and diamonds (♦). Each suit has 13 denominations, or ranks, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (Jack), Q (Queen), K (King) and A (Ace). But sometimes, a card goes missing – lost between the couch cushions, clothes-pinned in the bike spokes, filched by the cat, eaten by the dog (see Figure 1), or maybe misplaced in a matching deck. You may find yourself not playing with a full deck. Building a house of cards? No problem. Playing poker? Problem. Or is it? Ask yourself the following question:

What is the probability of being dealt a Two-Pair hand of five cards from a well-shuffled deck when the Ace of Spades is missing from the deck? Specifically, how does it compare to the probability of being dealt a Two-Pair hand from a standard 52-card deck?

What does your ‘probability intuition’ tell you? On the one hand, with a missing Ace of Spades there are fewer Two-Pair hands. On the other hand, there are also fewer total five-card hands. The ratio of these two numbers defines the probability; which way it goes seems unclear. Let’s do the calculation – first for the standard 52-card deck and then for the 51-card deck missing the Ace of Spades.