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 $\square$ . $\newline$ (Round to six decimal places as needed.)

Full solution

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

\square

\newline

(Round to six decimal places as needed.)

Calculate Queen Ways: First, let's find out how many ways we can pick one queen out of the four available in the deck. $\newline$ There are $4$ queens in a deck, so there are $4$ ways to pick one queen. $\newline$ Calculation: $4C1 = 4$
Calculate King Ways: Next, we need to figure out how many ways we can pick three kings out of the four available in the deck. $\newline$ There are $4$ kings in a deck, so there are $4$ ways to pick three kings. $\newline$ Calculation: $4C3 = 4$
Calculate Total Ways: Now, we need to calculate the total number of ways to pick any $4$ cards out of $52$ . $\newline$ Calculation: $52C4 = \frac{52\times51\times50\times49}{4\times3\times2\times1}$
Calculate Probability: Let's calculate the probability of getting one queen and three kings. $\newline$ Probability = (Ways to pick one queen $\times$ Ways to pick three kings) / Total ways to pick $4$ cards $\newline$ Calculation: Probability = $\frac{4 \times 4}{\binom{52}{4}}$
Perform Math Calculation: Now, we do the actual math. $\newline$ Calculation: Probability = $(4 \times 4) / \left((52\times51\times50\times49)/(4\times3\times2\times1)\right)$
Simplify Calculation: Simplify the calculation. $\newline$ Calculation: Probability = $\frac{16}{270725}$
Convert to Decimal: Finally, we convert this to a decimal to get the probability. $\newline$ Calculation: Probability = $\frac{16}{270725} \approx 0.000059$