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 or a face card?

Explanation:

Step1: Determine total number of cards

There are 52 cards in a standard deck.

Step2: Calculate number of black cards

There are 26 black cards (13 spades and 13 clubs). So the probability of drawing a black card $P(B)=\frac{26}{52}=\frac{1}{2}$.

Step3: Calculate number of face - cards

There are 12 face - cards (4 Jacks, 4 Queens, 4 Kings). So the probability of drawing a face - card $P(F)=\frac{12}{52}=\frac{3}{13}$.

Step4: Calculate number of black face - cards

There are 6 black face - cards (2 Jacks, 2 Queens, 2 Kings of spades and clubs). So $P(B\cap F)=\frac{6}{52}=\frac{3}{26}$.

Step5: Use the addition rule for probability

The addition rule is $P(B\cup F)=P(B)+P(F)-P(B\cap F)$. Substitute the values: $P(B\cup F)=\frac{26}{52}+\frac{12}{52}-\frac{6}{52}=\frac{26 + 12-6}{52}=\frac{32}{52}=\frac{8}{13}$.

Answer:

(a) $\frac{1}{2}$
(b) $\frac{3}{13}$
(c) $\frac{8}{13}$