If you are dealt $4$ cards from a shuffled deck of $52$ cards, find the probability of getting one queen and three kings. $\newline$ The probability is $0.000059$ . $\newline$ (Round to six decimal places as needed.)

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Q. If you are dealt

4

cards from a shuffled deck of

52

cards, find the probability of getting one queen and three kings.

\newline

The probability is

0.000059

\newline

(Round to six decimal places as needed.)

Count Queens Draw: Determine the number of ways to draw one queen from the deck. $\newline$ There are $4$ queens in a deck of $52$ cards. So, the number of ways to draw one queen is $4$ .
Count Kings Draw: Determine the number of ways to draw three kings from the deck. $\newline$ There are $4$ kings in a deck of $52$ cards. We need to choose $3$ out of these $4$ kings. The number of ways to do this is given by the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number of items to choose from, $k$ is the number of items to choose, and $!$ denotes factorial. $\newline$ So, the number of ways to draw three kings is $C(4, 3) = \frac{4!}{3!(4-3)!} = 4$ .
Total Cards Draw: Calculate the total number of ways to draw $4$ cards from the deck. $\newline$ The total number of ways to draw $4$ cards from a deck of $52$ is given by the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$ . $\newline$ So, the total number of ways to draw $4$ cards is $C(52, 4) = \frac{52!}{4!(52-4)!} = \frac{52\times51\times50\times49}{4\times3\times2\times1} = 270,725$ .
Calculate Probability: Calculate the probability of drawing one queen and three kings. $\newline$ The probability is the number of favorable outcomes divided by the total number of possible outcomes. $\newline$ The number of favorable outcomes is the product of the number of ways to draw one queen and the number of ways to draw three kings, which is $4 \times 4 = 16$ . $\newline$ The probability is therefore $\frac{16}{270,725}$ .
Perform Division: Perform the division to find the probability. $\frac{16}{270,725} = 0.000059$ (rounded to six decimal places).