Question

you are dealt one card from a 52-card deck. find the probability that you are dealt a heart or a picture card. click the icon to view a description of a standard deck of playing cards. the probability is \boxed{}. (type an integer or a simplified fraction.)

Explanation:

Step1: Define Events and Counts

Let \( A \) be the event of drawing a heart, \( B \) be the event of drawing a picture card. A standard deck has 52 cards. Number of hearts (\( n(A) \)): 13 (hearts suit). Picture cards: Jack, Queen, King in each suit, so 3 per suit × 4 suits = 12, so \( n(B) = 12 \). Overlap (heart picture cards): 3 (Jack, Queen, King of hearts), so \( n(A \cap B) = 3 \).

Step2: Apply Inclusion - Exclusion Principle

The formula for \( P(A \cup B) \) is \( P(A) + P(B) - P(A \cap B) \). Calculate each probability: \( P(A)=\frac{13}{52} \), \( P(B)=\frac{12}{52} \), \( P(A \cap B)=\frac{3}{52} \).

Step3: Compute the Result

Substitute into the formula: \( \frac{13}{52}+\frac{12}{52}-\frac{3}{52}=\frac{13 + 12 - 3}{52}=\frac{22}{52}=\frac{11}{26} \).

Answer:

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