Question

1. there are 52 cards in a standard deck. 12 of those cards are face cards (4 jacks, 4 queens, and 4 kings). the deck is shuffled and a group of 4 cards is dealt.

the probability that all 4 cards are face cards is ______. round your answer to the nearest thousandth. (lesson 19.3) (1 point)

Explanation:

Step1: Calculate the number of ways to choose 4 face cards

The number of ways to choose \(r\) items from \(n\) items is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\).
Here, \(n = 12\) (number of face cards) and \(r=4\).
\(C(12,4)=\frac{12!}{4!(12 - 4)!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495\)

Step2: Calculate the number of ways to choose 4 cards from 52 cards

Here, \(n = 52\) and \(r = 4\)
\(C(52,4)=\frac{52!}{4!(52-4)!}=\frac{52\times51\times50\times49}{4\times3\times2\times1}=270725\)

Step3: Calculate the probability

The probability \(P\) that all 4 cards are face cards is the ratio of the number of ways to choose 4 face cards to the number of ways to choose 4 cards from 52 cards.
\(P=\frac{C(12,4)}{C(52,4)}=\frac{495}{270725}\approx0.00183\)

Answer:

\(0.002\) (rounded to the nearest thousandth)