Answer :
To determine the probability that a randomly drawn card from a well-shuffled deck of playing cards is either an ace or a red card, follow these detailed steps:
1. Understand the Total Deck Composition:
- A standard deck contains 52 playing cards.
2. Identify the Number of Aces:
- There are 4 aces in a standard deck (one from each suit: hearts, diamonds, clubs, and spades).
3. Identify the Number of Red Cards:
- There are 26 red cards in a deck (13 hearts and 13 diamonds).
4. Identify the Overlapping Cards (Red Aces):
- There are 2 red aces in the deck (the ace of hearts and the ace of diamonds).
5. Calculate the Favorable Outcomes:
- To avoid counting the red aces twice, use the principle of inclusion and exclusion:
- Total number of aces: 4
- Total number of red cards: 26
- Number of red aces (overlap): 2
- The formula to find the non-overlapping favorable outcomes is:
[tex]\[ \text{Favorable outcomes} = \text{Number of Aces} + \text{Number of Red Cards} - \text{Number of Red Aces} \][/tex]
- Substituting in the values:
[tex]\[ \text{Favorable outcomes} = 4 + 26 - 2 = 28 \][/tex]
6. Calculate the Probability:
- Probability is the number of favorable outcomes divided by the total number of possible outcomes (total cards in the deck):
[tex]\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total cards}} = \frac{28}{52} \][/tex]
7. Simplify the Fraction:
- Simplify [tex]\(\frac{28}{52}\)[/tex] to its lowest terms by finding the greatest common divisor (GCD) of 28 and 52, which is 4:
[tex]\[ \frac{28 \div 4}{52 \div 4} = \frac{7}{13} \][/tex]
Thus, the probability that a randomly drawn card from a well-shuffled deck is an ace or a red card is [tex]\(\frac{7}{13}\)[/tex].
1. Understand the Total Deck Composition:
- A standard deck contains 52 playing cards.
2. Identify the Number of Aces:
- There are 4 aces in a standard deck (one from each suit: hearts, diamonds, clubs, and spades).
3. Identify the Number of Red Cards:
- There are 26 red cards in a deck (13 hearts and 13 diamonds).
4. Identify the Overlapping Cards (Red Aces):
- There are 2 red aces in the deck (the ace of hearts and the ace of diamonds).
5. Calculate the Favorable Outcomes:
- To avoid counting the red aces twice, use the principle of inclusion and exclusion:
- Total number of aces: 4
- Total number of red cards: 26
- Number of red aces (overlap): 2
- The formula to find the non-overlapping favorable outcomes is:
[tex]\[ \text{Favorable outcomes} = \text{Number of Aces} + \text{Number of Red Cards} - \text{Number of Red Aces} \][/tex]
- Substituting in the values:
[tex]\[ \text{Favorable outcomes} = 4 + 26 - 2 = 28 \][/tex]
6. Calculate the Probability:
- Probability is the number of favorable outcomes divided by the total number of possible outcomes (total cards in the deck):
[tex]\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total cards}} = \frac{28}{52} \][/tex]
7. Simplify the Fraction:
- Simplify [tex]\(\frac{28}{52}\)[/tex] to its lowest terms by finding the greatest common divisor (GCD) of 28 and 52, which is 4:
[tex]\[ \frac{28 \div 4}{52 \div 4} = \frac{7}{13} \][/tex]
Thus, the probability that a randomly drawn card from a well-shuffled deck is an ace or a red card is [tex]\(\frac{7}{13}\)[/tex].
