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Worked solution

Condition on the thirteen spades

A card is picked at random from a deck of 52 playing cards. Given that it is a spade, find the probability that it is a picture card.

CJC · Lecture · Example 2

Define the events and count their overlap

Let A mean picture card and B mean spade. Picture cards are J,Q,K. Thus B contains 13 cards and A∩BA\cap B contains J♠,Q♠,K♠:3 cards.

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Method 1: use the conditional probability formula

The source recommends the probability formula: P(A∣B)=P(A∩B)/P(B)=(3/52)/(13/52)=3/13P(A\mid B)=P(A\cap B)/P(B)=(3/52)/(13/52)=3/13.

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Method 2: reduce the sample space

Given B, the 13 spades are equally likely. There are 3 favourable cards, so P(A∣B)=n(A∩B)/n(B)=3/13P(A\mid B)=n(A\cap B)/n(B)=3/13. This counting shortcut requires equally likely outcomes within the reduced space; it is convenient here because that space can be listed easily.

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Source deck reconstructed as four rows Clubs, Diamonds, Hearts, Spades, each with ranks Ace,2,3,4,5,6,7,8,9,10, Jack, Queen, King. The last three columns J,Q,K are picture cards. Thus thirteen spades include three picture cards. The original decorative card-face artwork is replaced by accessible suit/rank labels without changing the inventory.
Source deck reconstructed as four rows Clubs, Diamonds, Hearts, Spades, each with ranks Ace,2,3,4,5,6,7,8,9,10, Jack, Queen, King. The last three columns J,Q,K are picture cards. Thus thirteen spades include three picture cards. The original decorative card-face artwork is replaced by accessible suit/rank labels without changing the inventory.
Source deck reconstructed as four rows Clubs, Diamonds, Hearts, Spades, each with ranks Ace,2,3,4,5,6,7,8,9,10, Jack, Queen, King. The last three columns J,Q,K are picture cards. Thus thirteen spades include three picture cards. The original decorative card-face artwork is replaced by accessible suit/rank labels without changing the inventory.

Common mistakes

Use all 13 spades as the conditional denominator. The Ace is not a picture card; only J,Q,K count. Uniformly drawing from a standard 52 carddeck supplies the equal-likelihood assumption.

Solution notes

The source card illustration creditshttp://math.hws.edu/eck/cs124/s13/lab13/index.html. Native suit/rank labels preserve all 52 card identities and the three picture-card columns; decorative card-face artwork is not needed for the count.

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