Question

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

A standard deck has 52 cards.

Step2: Calculate number of black cards

There are 26 black cards (13 spades and 13 clubs), so \(P(\text{black})=\frac{26}{52}=\frac{1}{2}\).

Step3: Calculate number of face - cards

Face - cards are Jacks (J), Queens (Q), and Kings (K). There are 3 face - cards in each of the 4 suits, so there are 12 face - cards. \(P(\text{face})=\frac{12}{52}=\frac{3}{13}\).

Step4: Calculate number of black face - cards

There are 6 black face - cards (3 in spades and 3 in clubs).

Step5: Use the addition rule of probability

The addition rule for two events \(A\) and \(B\) is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Let \(A\) be the event of drawing a black card and \(B\) be the event of drawing a face - card. Then \(P(A\cup B)=\frac{26}{52}+\frac{12}{52}-\frac{6}{52}\).
\[

$$\begin{align*} P(A\cup B)&=\frac{26 + 12-6}{52}\\ &=\frac{32}{52}\\ &=\frac{8}{13} \end{align*}$$

\]

Answer:

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