What is the probability of getting a King or a Heart greater than 10 expressed as a fraction?
Understand the Problem
The question is asking for the probability of drawing either a King or a Heart from a standard deck of cards, particularly focusing on Hearts that have a value greater than 10, which includes the Jack, Queen, and King of Hearts. We need to determine how many favorable outcomes exist and divide that number by the total number of cards in the deck.
Answer
The probability is $\frac{3}{26}$.
Answer for screen readers
The probability of drawing either a King or a Heart that is greater than 10 is $\frac{3}{26}$.
Steps to Solve
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Identify Total Number of Cards A standard deck of cards consists of 52 cards.
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Count Favorable Outcomes for Drawing a King There are 4 Kings in a deck (one from each suit: Hearts, Diamonds, Clubs, Spades).
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Count Favorable Outcomes for Drawing a Heart Greater than 10 The Hearts greater than 10 are the Jack, Queen, and King of Hearts. This gives us 3 favorable outcomes.
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Avoid Double Counting Since the King of Hearts is counted in both favorable outcomes (King and Heart), we need to ensure we count it only once. Thus, we count:
- Total Kings: 4
- Total Hearts greater than 10: 3
- The King of Hearts has been counted in both categories, so we subtract it once.
- Calculate Total Favorable Outcomes Using the principle of inclusion-exclusion: Total Favorable Outcomes = (Total Kings) + (Hearts greater than 10) - (King of Hearts)
This gives: $$ 4 + 3 - 1 = 6 $$
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Calculate Probability Now, we divide the total favorable outcomes by the total number of cards: $$ P = \frac{6}{52} $$
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Simplify the Probability To simplify, divide both the numerator and the denominator by 2: $$ P = \frac{3}{26} $$
The probability of drawing either a King or a Heart that is greater than 10 is $\frac{3}{26}$.
More Information
In a standard deck, there are 52 cards, and the calculation considers both the Kings and the Hearts above 10, accounting for the King of Hearts only once to avoid duplication.
Tips
- Forgetting to subtract the King of Hearts when counting both Kings and Hearts, which may lead to double counting the King of Hearts.
- Not simplifying the final fraction, which can make the answer appear less precise.
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