\begin{tabular}{l} Question 3 (Multiple Choice Worth 6 points) \\ (05.01 LC) \\ Assume you are working with a standard deck of 52 cards. There are 13 cards \( (2,3,4,5,6,7,8,9,10 \), jack, queen, king, and ace) in each of \\ four suits (clubs, diamonds, hearts, and spades). \\ What is P(Ace u Heart) if you draw one card? \\ \( \frac{1}{52} \) \\ \( \frac{4}{13} \) \\ \( \frac{17}{52} \) \\ \hline\end{tabular}

To find \( P(Ace \cup Heart) \), we need to determine the probability of drawing either an Ace or a Heart from a standard deck of 52 cards. ### Step 1: Identify the total number of outcomes The total number of cards in a standard deck is 52. ### Step 2: Identify the number of favorable outcomes 1. **Number of Aces**: There are 4 Aces in the deck (one for each suit). 2. **Number of Hearts**: There are 13 Hearts in the deck. ### Step 3: Identify the overlap Since one of the Aces is also a Heart (the Ace of Hearts), we need to account for this overlap when calculating the total number of favorable outcomes. ### Step 4: Use the formula for the union of two events The formula for the probability of the union of two events is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Where: - \( P(A) \) is the probability of drawing an Ace. - \( P(B) \) is the probability of drawing a Heart. - \( P(A \cap B) \) is the probability of drawing the Ace of Hearts. ### Step 5: Calculate each probability - \( P(A) = \frac{4}{52} \) (4 Aces) - \( P(B) = \frac{13}{52} \) (13 Hearts) - \( P(A \cap B) = \frac{1}{52} \) (1 Ace of Hearts) ### Step 6: Substitute into the formula Now we can substitute these values into the formula: \[ P(A \cup B) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} \] ### Step 7: Simplify the expression \[ P(A \cup B) = \frac{4 + 13 - 1}{52} = \frac{16}{52} \] ### Step 8: Reduce the fraction \[ P(A \cup B) = \frac{4}{13} \] ### Final Answer Thus, the probability \( P(Ace \cup Heart) \) is \( \frac{4}{13} \). The correct answer is \( \frac{4}{13} \).