Question

if you are dealt 5 cards from a shuffled deck of 52 cards, find the probability of getting two queens and three kings. click the icon to view a description of a standard deck of playing cards. the probability is (round to six decimal places as needed.)

Explanation:

Step1: Calculate number of ways to choose 2 queens

There are 4 queens in a deck. The number of ways to choose 2 queens out of 4 is given by the combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 4$ and $k=2$. So $C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4!}{2!2!}=\frac{4\times3}{2\times1}=6$.

Step2: Calculate number of ways to choose 3 kings

There are 4 kings in a deck. The number of ways to choose 3 kings out of 4 is $C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=4$.

Step3: Calculate number of favorable hands

The number of favorable hands (2 queens and 3 kings) is the product of the number of ways to choose 2 queens and the number of ways to choose 3 kings, so $6\times4 = 24$.

Step4: Calculate total number of 5 - card hands

The total number of 5 - card hands from a 52 - card deck is $C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960$.

Step5: Calculate probability

The probability $P$ is the number of favorable hands divided by the total number of hands, so $P=\frac{24}{2598960}\approx0.000009$.

Answer:

$0.000009$