Question

from a 52-card deck, what is p(draw a heart or a face card)? show use of general addition rule and compute exact simplified fraction.

Explanation:

Step1: Recall General Addition Rule

The general 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 heart and \( B \) be the event of drawing a face card.

Step2: Find \( P(A) \)

In a 52 - card deck, there are 13 hearts. So \( P(A)=\frac{13}{52}=\frac{1}{4} \).

Step3: Find \( P(B) \)

Face cards are Jack, Queen, King. There are 3 face cards per suit and 4 suits, so \( 3\times4 = 12 \) face cards. Thus \( P(B)=\frac{12}{52}=\frac{3}{13} \).

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

The intersection of hearts and face cards: there are 3 face cards in hearts (Jack, Queen, King of hearts). So \( P(A \cap B)=\frac{3}{52} \).

Step5: Apply General Addition Rule

\( P(A \cup B)=P(A)+P(B)-P(A \cap B)=\frac{13}{52}+\frac{12}{52}-\frac{3}{52}=\frac{13 + 12-3}{52}=\frac{22}{52}=\frac{11}{26} \).

Answer:

\(\frac{11}{26}\)