Four cards are drawn from a pack of cards. Find the probability that (i) all are diamonds (ii) there are two spades and two hearts.
Understand the Problem
The question asks to calculate the probability of two different events when drawing four cards from a standard deck: (i) all four cards are diamonds and (ii) two cards are spades and two cards are hearts. We need to use probability principles and combinatorics (combinations) to solve each part.
Answer
(i) $\frac{11}{4165}$ (ii) $\frac{468}{20825}$
Answer for screen readers
(i) The probability of drawing four diamonds is $\frac{11}{4165}$ (ii) The probability of drawing two spades and two hearts is $\frac{468}{20825}$
Steps to Solve
- Calculate the total number of ways to draw 4 cards from a deck of 52
The total number of ways to choose 4 cards from 52 is given by the combination formula:
$$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$
where $n$ is the total number of items, and $k$ is the number of items to choose. In our case, $n = 52$ and $k = 4$, so we have:
$$ \binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52!}{4!48!} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 $$
- Calculate the number of ways to draw 4 diamonds
There are 13 diamonds in a deck of cards. The number of ways to choose 4 diamonds from 13 is:
$$ \binom{13}{4} = \frac{13!}{4!(13-4)!} = \frac{13!}{4!9!} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 715 $$
- Calculate the probability of drawing 4 diamonds
The probability is the number of ways to draw 4 diamonds divided by the total number of ways to draw 4 cards:
$$ P(\text{4 diamonds}) = \frac{\binom{13}{4}}{\binom{52}{4}} = \frac{715}{270725} = \frac{11}{4165} $$
- Calculate the number of ways to draw 2 spades and 2 hearts
There are 13 spades and 13 hearts in a deck of cards. The number of ways to choose 2 spades from 13 is:
$$ \binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = 78 $$
Similarly, the number of ways to choose 2 hearts from 13 is:
$$ \binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = 78 $$
To get the number of ways to draw 2 spades and 2 hearts, we multiply these two results:
$$ \binom{13}{2} \times \binom{13}{2} = 78 \times 78 = 6084 $$
- Calculate the probability of drawing 2 spades and 2 hearts
The probability is the number of ways to draw 2 spades and 2 hearts divided by the total number of ways to draw 4 cards:
$$ P(\text{2 spades and 2 hearts}) = \frac{\binom{13}{2} \times \binom{13}{2}}{\binom{52}{4}} = \frac{6084}{270725} = \frac{468}{20825} $$
(i) The probability of drawing four diamonds is $\frac{11}{4165}$ (ii) The probability of drawing two spades and two hearts is $\frac{468}{20825}$
More Information
These probabilities are relatively low due to the large number of possible combinations when drawing four cards from a standard deck of 52 cards.
Tips
A common mistake is calculating permutations instead of combinations. Since the order in which the cards are drawn does not matter, we should use combinations. Another common mistake involves not correctly calculating $n!$ - make sure to carry out the calculations correctly.
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