Alright, let's solve the problem step-by-step.
1. Identify the Total Number of Cards:
- A standard deck of playing cards has 52 cards.
2. Identify the Number of Aces:
- There are 4 aces in a deck (Ace of Spades, Ace of Hearts, Ace of Clubs, and Ace of Diamonds).
3. Identify the Number of Red Cards:
- There are 26 red cards in a deck (13 Hearts and 13 Diamonds).
4. Identify the Number of Red Aces:
- Out of the 4 aces, 2 are red (Ace of Hearts and Ace of Diamonds).
5. Use the Formula for the Probability of an Event (P(Ace or Red Card)):
- To calculate the probability of drawing a card that is either an Ace or a Red card, we use the formula for the union of two events:
[tex]\[
P(A \cup R) = P(A) + P(R) - P(A \cap R)
\][/tex]
- Where:
- [tex]\( P(A) \)[/tex] = Probability of drawing an Ace.
- [tex]\( P(R) \)[/tex] = Probability of drawing a Red Card.
- [tex]\( P(A \cap R) \)[/tex] = Probability of drawing a card that is both an Ace and a Red Card.
6. Calculate Each Probability:
- [tex]\( P(A) \)[/tex] = Number of Aces / Total number of cards = [tex]\( \frac{4}{52} \)[/tex]
- [tex]\( P(R) \)[/tex] = Number of Red Cards / Total number of cards = [tex]\( \frac{26}{52} \)[/tex]
- [tex]\( P(A \cap R) \)[/tex] = Number of Red Aces / Total number of cards = [tex]\( \frac{2}{52} \)[/tex]
7. Combine the Probabilities:
- Plugging the values into our formula:
[tex]\[
P(A \cup R) = \frac{4}{52} + \frac{26}{52} - \frac{2}{52}
\][/tex]
- Simplify the fractions:
[tex]\[
P(A \cup R) = \frac{4 + 26 - 2}{52} = \frac{28}{52}
\][/tex]
8. Simplify the Final Answer:
- To reduce the fraction [tex]\( \frac{28}{52} \)[/tex], find the greatest common divisor (GCD) of 28 and 52, which is 4:
[tex]\[
\frac{28}{52} = \frac{28 \div 4}{52 \div 4} = \frac{7}{13}
\][/tex]
Therefore, the probability that the card drawn is an Ace or a Red card is [tex]\( \frac{7}{13} \)[/tex].
