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: Calculate the number of face cards

In a standard 52 - card deck, there are 12 face cards (4 Jacks, 4 Queens, 4 Kings).

Step2: Calculate the probability of drawing a face card

The probability formula is \(P(A)=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of favorable outcomes and \(n(S)\) is the total number of outcomes. Here, \(n(A) = 12\) and \(n(S)=52\). So \(P(\text{face card})=\frac{12}{52}=\frac{3}{13}\).

Step3: Calculate the probability of not drawing a face card

We use the formula \(P(\text{not }A)=1 - P(A)\). So \(P(\text{not face card})=1-\frac{3}{13}=\frac{13 - 3}{13}=\frac{10}{13}\).

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

The odds in favor formula is \(\text{Odds in favor}=\frac{P(A)}{P(\text{not }A)}\). Substituting \(P(A)=\frac{3}{13}\) and \(P(\text{not }A)=\frac{10}{13}\), we get \(\frac{\frac{3}{13}}{\frac{10}{13}}=\frac{3}{10}\), or \(3:10\).

Step5: Calculate the odds against drawing a face card

The odds against formula is \(\text{Odds against}=\frac{P(\text{not }A)}{P(A)}\). Substituting \(P(A)=\frac{3}{13}\) and \(P(\text{not }A)=\frac{10}{13}\), we get \(\frac{\frac{10}{13}}{\frac{3}{13}}=\frac{10}{3}\), or \(10:3\).

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: \(3:10\)
Odds against drawing a face card: \(10:3\)