Questions: A card is drawn randomly from a standard 52-card deck. Find the following: Write all answers as simplified fractions (not mixed numbers). Probability the card drawn is a face card Probability the card drawn is not a face card. Odds in favor of drawing a face card Odds against drawing a face card

$A card is drawn randomly from a standard 52-card deck. Find the following: Write all answers as simplified fractions (not mixed numbers). Probability the card drawn is a face card Probability the card drawn is not a face card. Odds in favor of drawing a face card Odds against drawing a face card$

Transcript text: A card is drawn randomly from a standard 52 -card deck. Find the following: Write all answers as simplified fractions (not mixed numbers). Probability the card drawn is a face card $\square$ Probability the card drawn is not a face card. $\square$ Odds in favor of drawing a face card $\square$ Odds against drawing a face card $\square$

Solution

Solution Steps

To solve the given problem, we need to understand the composition of a standard 52-card deck and the definitions of probability and odds.

Probability the card drawn is a face card: There are 12 face cards in a deck (4 Jacks, 4 Queens, 4 Kings). The probability is the number of face cards divided by the total number of cards.
Probability the card drawn is not a face card: This is the complement of the probability of drawing a face card.
Odds in favor of drawing a face card: This is the ratio of the number of face cards to the number of non-face cards.
Odds against drawing a face card: This is the ratio of the number of non-face cards to the number of face cards.

Step 1: Probability the Card Drawn is a Face Card

The probability $ P(\text{face card}) $ of drawing a face card from a standard 52-card deck is calculated as follows: \[ P(\text{face card}) = \frac{\text{Number of face cards}}{\text{Total number of cards}} = \frac{12}{52} = \frac{3}{13} \approx 0.2308 \]

Step 2: Probability the Card Drawn is Not a Face Card

The probability $ P(\text{not face card}) $ of drawing a card that is not a face card is the complement of the probability of drawing a face card: \[ P(\text{not face card}) = 1 - P(\text{face card}) = 1 - \frac{3}{13} = \frac{10}{13} \approx 0.7692 \]

Step 3: Odds in Favor of Drawing a Face Card

The odds in favor of drawing a face card are given by the ratio of the number of face cards to the number of non-face cards: \[ \text{Odds in favor} = \frac{\text{Number of face cards}}{\text{Number of non-face cards}} = \frac{12}{40} = \frac{3}{10} = 0.3 \]

Step 4: Odds Against Drawing a Face Card

The odds against drawing a face card are given by the ratio of the number of non-face cards to the number of face cards: \[ \text{Odds against} = \frac{\text{Number of non-face cards}}{\text{Number of face cards}} = \frac{40}{12} = \frac{10}{3} \approx 3.3333 \]