Question

a poker hand consists of five cards randomly dealt from a standard deck of 52 cards. the order of the cards does not matter. determine the following probabilities for a 5 - card poker hand. write your answers in percent form, rounded to 4 decimal places. determine the probability that exactly 3 of these cards are aces. answer: % determine the probability that all five of these cards are spades. answer: % determine the probability that exactly 3 of these cards are face cards. answer: % determine the probability of selecting exactly 2 aces and exactly 2 kings. answer: % determine the probability of selecting exactly 1 jack. answer: %

Explanation:

Step1: Calculate total number of 5 - card hands

The total number of 5 - card hands from a 52 - card deck is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 52$ and $r=5$. So, $C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960$.

Step2: Probability of exactly 3 Aces

There are 4 Aces in a deck. The number of ways to choose 3 Aces out of 4 is $C(4,3)=\frac{4!}{3!(4 - 3)!}=4$. The number of ways to choose the remaining $5-3 = 2$ non - Aces out of $52 - 4=48$ non - Aces is $C(48,2)=\frac{48!}{2!(48 - 2)!}=\frac{48\times47}{2\times1}=1128$. The number of hands with exactly 3 Aces is $C(4,3)\times C(48,2)=4\times1128 = 4512$. The probability is $\frac{4512}{2598960}\times100\%\approx0.1736\%$.

Step3: Probability of all 5 Spades

There are 13 Spades in a deck. The number of ways to choose 5 Spades out of 13 is $C(13,5)=\frac{13!}{5!(13 - 5)!}=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}=1287$. The probability is $\frac{1287}{2598960}\times100\%\approx0.0495\%$.

Step4: Probability of exactly 3 face cards

There are 12 face cards in a deck. The number of ways to choose 3 face cards out of 12 is $C(12,3)=\frac{12!}{3!(12 - 3)!}=\frac{12\times11\times10}{3\times2\times1}=220$. The number of ways to choose the remaining $5 - 3=2$ non - face cards out of $52 - 12 = 40$ non - face cards is $C(40,2)=\frac{40!}{2!(40 - 2)!}=\frac{40\times39}{2\times1}=780$. The number of hands with exactly 3 face cards is $C(12,3)\times C(40,2)=220\times780 = 171600$. The probability is $\frac{171600}{2598960}\times100\%\approx6.6025\%$.

Step5: Probability of exactly 2 Aces and 2 Kings

There are 4 Aces and 4 Kings. The number of ways to choose 2 Aces out of 4 is $C(4,2)=\frac{4!}{2!(4 - 2)!}=6$. The number of ways to choose 2 Kings out of 4 is $C(4,2)=6$. The number of ways to choose the remaining $5-(2 + 2)=1$ card out of $52-(4 + 4)=44$ cards is $C(44,1)=44$. The number of hands with exactly 2 Aces and 2 Kings is $C(4,2)\times C(4,2)\times C(44,1)=6\times6\times44 = 1584$. The probability is $\frac{1584}{2598960}\times100\%\approx0.0609\%$.

Step6: Probability of exactly 1 Jack

There are 4 Jacks. The number of ways to choose 1 Jack out of 4 is $C(4,1)=4$. The number of ways to choose the remaining $5 - 1=4$ non - Jack cards out of $52 - 4 = 48$ non - Jack cards is $C(48,4)=\frac{48!}{4!(48 - 4)!}=\frac{48\times47\times46\times45}{4\times3\times2\times1}=194580$. The number of hands with exactly 1 Jack is $C(4,1)\times C(48,4)=4\times194580 = 778320$. The probability is $\frac{778320}{2598960}\times100\%\approx29.9460\%$.

Answer:

0.1736%
0.0495%
6.6025%
0.0609%
29.9460%