Question

there are 4 jacks and 13 clubs in a standard, 52 - card deck of playing cards. what is the probability that a card picked at random from a standard deck of playing cards is a club or a jack?
a. $\frac{1}{52}$
b. $\frac{3}{13}$
c. $\frac{4}{13}$
d. $\frac{7}{26}$

Explanation:

Step1: Recall the formula for probability of union

The formula for \( P(A \cup B) \) is \( P(A) + P(B) - P(A \cap B) \), where \( A \) is the event of picking a club and \( B \) is the event of picking a jack.

Step2: Calculate \( P(A) \)

There are 13 clubs in a 52 - card deck, so \( P(A)=\frac{13}{52} \).

Step3: Calculate \( P(B) \)

There are 4 jacks in a 52 - card deck, so \( P(B)=\frac{4}{52} \).

Step4: Calculate \( P(A \cap B) \)

The intersection of clubs and jacks is the jack of clubs, so there is 1 such card. Thus, \( P(A \cap B)=\frac{1}{52} \).

Step5: Calculate \( P(A \cup B) \)

Substitute the values into the formula: \( P(A \cup B)=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{13 + 4-1}{52}=\frac{16}{52}=\frac{4}{13} \).

Answer:

C. \(\frac{4}{13}\)