Math Problem Statement

A card is drawn randomly from a standard 52-card deck. Find the following: Write all answers as simplified fractions. Odds against drawing a face card.

Solution

A standard 52-card deck has 12 face cards: 4 Jacks, 4 Queens, and 4 Kings. The total number of cards is 52.

Step 1: Calculate the probability of drawing a face card

The probability of drawing a face card is:

P(Face card)=1252=313P(\text{Face card}) = \frac{12}{52} = \frac{3}{13}

Step 2: Calculate the probability of not drawing a face card

The probability of not drawing a face card is:

P(Not a face card)=1P(Face card)=1313=1013P(\text{Not a face card}) = 1 - P(\text{Face card}) = 1 - \frac{3}{13} = \frac{10}{13}

Step 3: Calculate the odds against drawing a face card

The odds against drawing a face card is the ratio of the probability of not drawing a face card to the probability of drawing a face card:

Odds against drawing a face card=P(Not a face card)P(Face card)=1013313=103\text{Odds against drawing a face card} = \frac{P(\text{Not a face card})}{P(\text{Face card})} = \frac{\frac{10}{13}}{\frac{3}{13}} = \frac{10}{3}

So, the odds against drawing a face card are 103\boxed{\frac{10}{3}}.


Do you want more details or have any questions?

Here are 5 related questions for further practice:

  1. What are the odds against drawing a King specifically?
  2. What is the probability of drawing a red card?
  3. What are the odds in favor of drawing a face card?
  4. What is the probability of drawing a non-face card?
  5. How many total black face cards are in the deck?

Tip: When calculating odds, always express them as a ratio between two probabilities—favoring an outcome vs. against that outcome.

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Math Problem Analysis

Mathematical Concepts

Probability
Odds

Formulas

P(Face card) = 12/52 = 3/13
P(Not a face card) = 1 - P(Face card) = 10/13
Odds against drawing a face card = P(Not a face card) / P(Face card) = 10/3

Theorems

Basic probability principles
Odds calculation formula

Suitable Grade Level

Grades 7-9