Question

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

Explanation:

Step1: Identify number of face - cards and total cards

A standard deck has 52 cards. Face - cards are Jacks, Queens, and Kings. There are 4 Jacks, 4 Queens, and 4 Kings, so there are 12 face - cards.

Step2: Calculate probability of drawing a face - card

The probability $P(F)$ of an event $F$ is given by $P(F)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. So, $P(\text{face - card})=\frac{12}{52}=\frac{3}{13}$.

Step3: Calculate probability of not drawing a face - card

The probability of the complement of an event $F$, denoted as $\overline{F}$, is $P(\overline{F}) = 1 - P(F)$. So, $P(\text{not face - card})=1-\frac{3}{13}=\frac{10}{13}$.

Step4: Calculate odds in favor of drawing a face - card

The odds in favor of an event $F$ is given by $\frac{P(F)}{P(\overline{F})}$. So, odds in favor $=\frac{\frac{3}{13}}{\frac{10}{13}}=\frac{3}{10}$.

Step5: Calculate odds against drawing a face - card

The odds against an event $F$ is the reciprocal of the odds in favor. So, odds against $=\frac{10}{3}$.

Answer:

Probability the card drawn is a face card: $\frac{3}{13}$
Probability the card drawn is not a face card: $\frac{10}{13}$
Odds in favor of drawing a face card: $\frac{3}{10}$
Odds against drawing a face card: $\frac{10}{3}$