Question

if you are dealt 4 cards from a shuffled deck of 52 cards, find the probability of getting three queens and one king. click the icon to view a description of a standard deck of playing cards. the probability is \\(\square\\). (round to six decimal places as needed.)

Explanation:

Step1: Calculate number of ways to choose 3 queens

There are 4 queens in a deck. The number of ways to choose 3 queens is given by the combination formula \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 4 \) (number of queens) and \( k=3 \) (number of queens we want to choose). So \( C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=\frac{4\times3!}{3!×1}=4 \)

Step2: Calculate number of ways to choose 1 king

There are 4 kings in a deck. The number of ways to choose 1 king is \( C(4,1)=\frac{4!}{1!(4 - 1)!}=\frac{4!}{1!3!}=\frac{4\times3!}{1\times3!}=4 \)

Step3: Calculate number of ways to choose 4 cards from 52

The number of ways to choose 4 cards from 52 is \( C(52,4)=\frac{52!}{4!(52 - 4)!}=\frac{52!}{4!48!}=\frac{52\times51\times50\times49}{4\times3\times2\times1}=270725 \)

Step4: Calculate the number of favorable outcomes

The number of favorable outcomes (3 queens and 1 king) is the product of the number of ways to choose 3 queens and the number of ways to choose 1 king. So \( 4\times4 = 16 \)

Step5: Calculate the probability

The probability \( P \) is the number of favorable outcomes divided by the number of total outcomes. So \( P=\frac{16}{270725}\approx0.000059 \)

Answer:

\( 0.000059 \)