Question

question 5
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 the total number of 5 - card hands

The total number of ways to choose 5 cards from a deck of 52 cards 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 that exactly 3 of the cards are Aces

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

Step3: Probability that all five of the cards are Spades

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

Step4: Probability that exactly 3 of the cards are face cards

There are 12 face cards (4 Jacks, 4 Queens, 4 Kings) in a deck. The number of ways to choose 3 face cards 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 2 non - face cards is \(C(40,2)=\frac{40!}{2!(40 - 2)!}=\frac{40\times39}{2\times1}=780\). The number of 5 - card hands with exactly 3 face cards is \(C(12,3)\times C(40,2)=220\times780 = 171600\). The probability \(P=\frac{171600}{2598960}\approx6.6028\%\)

Step5: Probability of selecting exactly 2 Aces and exactly 2 Kings

The number of ways to choose 2 Aces is \(C(4,2)=\frac{4!}{2!(4 - 2)!}=6\). The number of ways to choose 2 Kings is \(C(4,2)=6\). The number of ways to choose the remaining 1 non - Ace and non - King card is \(C(44,1)=44\). The number of 5 - card 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 \(P=\frac{1584}{2598960}\approx0.061\%\)

Step6: Probability of selecting exactly 1 Jack

There are 4 Jacks in a deck. The number of ways to choose 1 Jack is \(C(4,1)=4\). The number of ways to choose the remaining 4 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 5 - card hands with exactly 1 Jack is \(C(4,1)\times C(48,4)=4\times194580 = 778320\). The probability \(P=\frac{778320}{2598960}\approx29.95\%\)

Answer:

• Probability that exactly 3 of the cards are Aces: \(0.1736\%\)
• Probability that all five of the cards are Spades: \(0.0495\%\)
• Probability that exactly 3 of the cards are face cards: \(6.6028\%\)
• Probability of selecting exactly 2 Aces and exactly 2 Kings: \(0.061\%\)
• Probability of selecting exactly 1 Jack: \(29.95\%\)