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 five? (c) what is the probability that the card drawn is a black card or a five?

Explanation:

Step1: Calculate probability for (a)

Probability formula: \(P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
Total cards \(n = 52\). Black cards: \(26\) (spades and clubs).
\(P(\text{black})=\frac{26}{52}=\frac{1}{2}\).

Step2: Calculate probability for (b)

Fives: \(4\) (one in each suit).
\(P(\text{five})=\frac{4}{52}=\frac{1}{13}\).

Step3: Calculate probability for (c)

Use formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
\(P(\text{black})=\frac{26}{52}\), \(P(\text{five})=\frac{4}{52}\). Black - fives: \(2\) (five of spades and five of clubs). \(P(\text{black and five})=\frac{2}{52}\).
\(P(\text{black or five})=\frac{26 + 4- 2}{52}=\frac{28}{52}=\frac{7}{13}\).

Answer:

(a) \(\frac{1}{2}\)
(b) \(\frac{1}{13}\)
(c) \(\frac{7}{13}\)