You shuffle a standard deck of cards, then draw four cards. What is the probability all four are the same suit? What is the probability all four are red? What is the probability ea... You shuffle a standard deck of cards, then draw four cards. What is the probability all four are the same suit? What is the probability all four are red? What is the probability each has a different suit?
Understand the Problem
The question is asking to calculate probabilities related to drawing four cards from a shuffled standard deck. Specifically, it asks for the probability of drawing four cards of the same suit, four red cards, and four cards each of a different suit.
Answer
Same Suit: $\frac{2860}{270725} \approx 0.01056$ Four Red: $\frac{14950}{270725} \approx 0.05522$ Different Suits: $\frac{28561}{270725} \approx 0.10549$
Answer for screen readers
Probability of drawing four cards of the same suit: $\frac{2860}{270725} \approx 0.01056$
Probability of drawing four red cards: $\frac{14950}{270725} \approx 0.05522$
Probability of drawing four cards each of a different suit: $\frac{28561}{270725} \approx 0.10549$
Steps to Solve
- Define the sample space
The sample space is the total number of ways to choose 4 cards from a deck of 52 cards. This can be calculated using combinations:
$$ \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52!}{4!48!} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 $$
- Calculate the probability of drawing four cards of the same suit
There are 4 suits (hearts, diamonds, clubs, spades). For each suit, there are 13 cards. The number of ways to choose 4 cards from one suit is $\binom{13}{4}$. Since there are 4 suits, we multiply this by 4:
$$ 4 \times \binom{13}{4} = 4 \times \frac{13!}{4!9!} = 4 \times \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 4 \times 715 = 2860 $$
The probability of drawing four cards of the same suit is:
$$ P(\text{same suit}) = \frac{2860}{270725} = \frac{2860}{270725} \approx 0.01056 $$
- Calculate the probability of drawing four red cards
There are 26 red cards (13 hearts and 13 diamonds). The number of ways to choose 4 red cards from 26 is:
$$ \binom{26}{4} = \frac{26!}{4!22!} = \frac{26 \times 25 \times 24 \times 23}{4 \times 3 \times 2 \times 1} = 14950 $$
The probability of drawing four red cards is:
$$ P(\text{four red cards}) = \frac{14950}{270725} = \frac{14950}{270725} \approx 0.05522 $$
- Calculate the probability of drawing four cards each of a different suit
We need to choose one card from each suit. There are 13 cards in each suit. So, we have 13 choices for each of the four suits. The total number of ways to choose one card from each suit is:
$$ 13 \times 13 \times 13 \times 13 = 13^4 = 28561 $$
The probability of drawing four cards each of a different suit is:
$$ P(\text{different suits}) = \frac{28561}{270725} \approx 0.10549 $$
Probability of drawing four cards of the same suit: $\frac{2860}{270725} \approx 0.01056$
Probability of drawing four red cards: $\frac{14950}{270725} \approx 0.05522$
Probability of drawing four cards each of a different suit: $\frac{28561}{270725} \approx 0.10549$
More Information
These probabilities give us an understanding of how likely certain card combinations are when drawing from a standard deck. Drawing four cards of the same suit is relatively rare, while drawing four cards of different suits is more common.
Tips
- Forgetting to account for the number of suits when calculating the probability of drawing four cards of the same suit.
- Incorrectly calculating combinations or permutations.
- Not understanding the difference between combinations and permutations (order doesn't matter here, so we use combinations).
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