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Learning Outcomes
Students will be able to

• find the probability of a compound event involving a deck of playing cards
• use words or diagrams to represent the sample space for the compound event 
• define the mathematical terms below

Common Core State Standards: 7.SP.C.7a, 7.SP.C.8a

Vocabulary: Suits (cards), random, sample space, probability, compound probability

Materials: Per group of four to six students: deck of 52 playing cards; per student: pencil, Pick a Card worksheet 

Procedure
1. Introduction (5–10 minutes, whole group)
Distribute the decks of cards and probe to determine how familiar students are with them. As needed, give students time to explore the cards.

Establish that the decks contain 52 cards in four suits, each with 13 cards. Then, engage students in discussing these questions:

• Suppose you are going to pick one card at random. Do you have an equal chance of choosing any card?
• What is the sample space?
• What is the probability of choosing a heart?

2. Pick a Card (10 minutes, pairs)
Distribute pencils and copies of the worksheet.  

As students work with a partner, circulate to encourage anyone having difficulty to try one of these strategies:

• Jot down all the cards and mark those that fit the problem, such as:
• HA H2 … HK (for the heart cards ace, 2, … king)
• CA C2 … CK (for the clubs)
• Lay out the cards and put aside those that fit the problem.

When everyone is done, call the group together to share results. 

3. Video (5 minutes, whole group) 
Pose the following compound probability problem: Find the probability of randomly selecting a queen or a heart from a deck of cards. 

Give students a few minutes to work on the problem. Then, show them the video. Stop at 0:30 to review the use of:

After watching the video, ask:

• Did anyone solve the problem the way Mr. Long did? 
• What other ways did you find to solve the problem? 
• Why do you need to account for double-counting in this problem, but not in #5 from the worksheet?

4. Conclusion (5 minutes, whole group) 
Pose a couple of problems that require accounting for double-counting, such as the chance of drawing:

• a 7, 9, or club
• an even number or a diamond

Conclude by posing this challenge: Imagine you draw the queen of hearts from a full deck. You keep the card. What is the probability that the next card you draw will be a queen? That it will be a heart?

Encourage students to consider the size of the sample space and the number of successful outcomes once the queen of hearts is no longer in the deck.  

Activity Extension: Have students solve this problem: Four people are playing a card game (Hearts). Each player starts with 13 cards. In the first round, each player gives up (“plays”) a club; one of the cards played is the ace of clubs. What is the probability that a club or an ace remains in play?

This activity is based on work developed at TERC.