Each card in a standard deck of playing cards is unique and belongs to one of four suits: thirteen cards are clubs thirteen cards are diamonds thirteen cards are hearts thirteen cards are spades Suppose that Luisa randomly draws five cards without replacement. What is the probability that Luisa gets three diamonds and two hearts (in any order)? Choose $1$ answer: $\newline$ Choose $1$ answer:

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Find probabilities using the addition rule

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Q. Each card in a standard deck of playing cards is unique and belongs to one of four suits: thirteen cards are clubs thirteen cards are diamonds thirteen cards are hearts thirteen cards are spades Suppose that Luisa randomly draws five cards without replacement. What is the probability that Luisa gets three diamonds and two hearts (in any order)? Choose

1

answer:

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Choose

1

answer:

Identify total number of cards: Identify the total number of cards in a standard deck and the number of each suit. A standard deck has $52$ cards, with each suit (clubs, diamonds, hearts, spades) having $13$ cards each.
Calculate ways to choose diamonds: Calculate the number of ways to choose $2$ diamonds from $13$ diamonds. Using the combination formula, $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number of items to choose from, and $k$ is the number of items to choose, we get $\binom{13}{2} = \frac{13!}{2!(13-2)!} = 78$ .
Calculate ways to choose heart: Calculate the number of ways to choose $1$ heart from $13$ hearts. Using the same combination formula, $\binom{13}{1} = \frac{13!}{1!(13-1)!} = 13$ .
Calculate total ways to choose $3$ cards: Calculate the total number of ways to choose any $3$ cards from the $52$ cards in the deck. Using the combination formula, $\binom{52}{3} = \frac{52!}{3!(52-3)!} = 22100$ .
Calculate probability of drawing cards: Calculate the probability of drawing $2$ diamonds and $1$ heart. Multiply the number of ways to choose $2$ diamonds and $1$ heart and then divide by the total number of ways to choose any $3$ cards. $P = \frac{\binom{13}{2} \times \binom{13}{1}}{\binom{52}{3}} = \frac{78 \times 13}{22100} = \frac{1014}{22100} \approx 0.0459$ .