Question

five cards are drawn randomly from a standard deck of 52 cards. determine the probability that exactly 3 of these cards are aces. write your answer in decimal form, rounded to 5 decimal places.

Explanation:

Step1: Calculate total number of ways to draw 5 - card hands

The number of combinations of choosing \(n\) items from \(m\) items is given by the formula \(C(m,n)=\frac{m!}{n!(m - n)!}\). Here, \(m = 52\) and \(n=5\), so the total number of 5 - card hands is \(C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960\).

Step2: Calculate number of ways to choose 3 aces

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

Step3: Calculate number of ways to choose the remaining 2 non - aces

There are \(52-4 = 48\) non - aces. The number of ways to choose 2 non - aces out of 48 is \(C(48,2)=\frac{48!}{2!(48 - 2)!}=\frac{48\times47}{2\times1}=1128\).

Step4: Calculate number of 5 - card hands with exactly 3 aces

By the multiplication principle, the number of 5 - card hands with exactly 3 aces is the product of the number of ways to choose 3 aces and the number of ways to choose 2 non - aces. So, the number of 5 - card hands with exactly 3 aces is \(C(4,3)\times C(48,2)=4\times1128 = 4512\).

Step5: Calculate the probability

The probability \(P\) that exactly 3 of the 5 cards are aces is \(P=\frac{C(4,3)\times C(48,2)}{C(52,5)}=\frac{4512}{2598960}\approx0.00174\).

Answer:

0.00174