Question

here is a table showing all 52 cards in a standard deck. suppose one card is drawn at random from a standard deck. answer each part. write your answers as fractions. (a) what is the probability that the card drawn is a black card? (b) what is the probability that the card drawn is a face card? (c) what is the probability that the card drawn is a black card and a face card?

Explanation:

Step1: Calculate total number of cards

Total number of cards \( n(S)=52 \)

Step2: Calculate number of black cards

Number of black cards (Spades and Clubs) \( n(\text{Black}) = 26 \)
Probability of drawing a black card \( P(\text{Black})=\frac{n(\text{Black})}{n(S)}=\frac{26}{52}=\frac{1}{2} \)

Step3: Calculate number of face cards

Number of face cards (Jack, Queen, King in each suit) \( n(\text{Face})=12 \)
Probability of drawing a face card \( P(\text{Face})=\frac{n(\text{Face})}{n(S)}=\frac{12}{52}=\frac{3}{13} \)

Step4: Calculate number of black - face cards

Number of black - face cards (Jack, Queen, King in Spades and Clubs) \( n(\text{Black and Face}) = 6 \)
Probability of drawing a black and face card \( P(\text{Black and Face})=\frac{n(\text{Black and Face})}{n(S)}=\frac{6}{52}=\frac{3}{26} \)

Answer:

(a) \(\frac{1}{2}\)
(b) \(\frac{3}{13}\)
(c) \(\frac{3}{26}\)