**Middle School Number Sense Lesson 96: Probability with a Deck of Cards**

This concept is most helpful for gamblers and blackjack dealers (which are hopefully none of you). But it is a pretty fun concept that shows up at every level of number sense competition—elementary, middle, and high school. In middle school, it appeared

**5 times**this year, with a median placement at

**question # 72**.

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A standard deck of playing cards 52 cards, divided evenly into

**4 suits**(2 red and 2 black):

- Hearts
- Diamonds
- Spades
- Clubs

Each suit has

**13 cards**:

- 9 numbered cards (2, 3, 4, 5, 6, 7, 8, 9, 10)
- 3 face cards (Jack, Queen, King), and
- 1 Ace

As a reminder, the probability of an event (E) happening is:

**# favorable outcomes**

P(E) = -----------------------------------

# unfavorable outcomes

P(E) = -----------------------------------

# unfavorable outcomes

**Example 1: The probability of drawing a 5 from a standard deck of playing cards is ___**

- Numerator:
**4**(1 in each suit) - Denominator:
**52** - 4/52 =
**1/13**

**Example 2: What is the probability of picking a single card from a standard deck of 52 cards and drawing a red king? ___**

- Numerator:
**2**(1 heart + 1 diamond) - Denominator:
**52** - 2/52 =
**1/26**

**Example 3: What is the probability of picking a single card from a standard deck of 52 cards and drawing a face card? ___**

- Numerator:
**12**(3 per suit) - Denominator:
**52** - 12/52 =
**3/13**

**Example 4: Find the probability of drawing a heart from a standard deck of 52 cards. ___**

- Numerator:
**13** - Denominator:
**52** - 13/52 =
**1/4**

**Example 5: What is the probability of drawing 2 consecutive hearts from a standard 52-card deck without replacement? ___**

- We have to break this one down into two pieces.
- First heart:
**1/4**(see Example 4) - Second heart Numerator:
**12**hearts left - Second heart Denominator:
**51**cards left - 12/51 =
**4/17** - Now we multiply to combine the results. 1/4 x 4/17 =
**1/17**

**Up Next for Middle School: RemainDiv3**