Question

if you are dealt 3 cards from a shuffled deck of 52 cards, find the probability of getting two queens and one king
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 the 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 \(k=2\).

$$C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3\times2!}{2!\times2!}=\frac{4\times3}{2\times 1}=6$$

Step2: Calculate the number of ways to choose 1 king

There are 4 kings in a deck. The number of ways to choose 1 king out of 4 is \(C(4,1)=\frac{4!}{1!(4 - 1)!}=\frac{4!}{1!\times3!}=4\)

Step3: Calculate the number of ways to choose 3 cards from 52 cards

The number of ways to choose 3 cards from 52 cards is \(C(52,3)=\frac{52!}{3!(52 - 3)!}=\frac{52\times51\times50\times49!}{3!\times49!}=\frac{52\times51\times50}{3\times2\times1}=22100\)

Step4: Calculate the probability

The probability \(P\) of getting two queens and one king is the product of the number of ways to choose 2 queens and 1 king divided by the number of ways to choose 3 cards from 52 cards.

$$P=\frac{C(4,2)\times C(4,1)}{C(52,3)}=\frac{6\times4}{22100}=\frac{24}{22100}\approx0.001086$$

Answer:

\(0.001086\)