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 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 given by the combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 4$ and $k=3$. So $C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=\frac{4\times3!}{3!}=4$.

Step2: Calculate number of ways to choose remaining 2 non - aces

There are $52-4 = 48$ non - aces. The number of ways to choose 2 non - aces out of 48 is $C(48,2)=\frac{48!}{2!(48 - 2)!}=\frac{48\times47\times46!}{2\times1\times46!}=1128$.

Step3: Calculate number of ways to choose 5 cards from 52

The number of ways to choose 5 cards from 52 is $C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48!}{5\times4\times3\times2\times1\times48!}=2598960$.

Step4: Calculate probability

The probability $P$ that exactly 3 of the 5 cards are aces is the product of the number of ways to choose 3 aces and 2 non - aces divided by the number of ways to choose 5 cards. So $P=\frac{C(4,3)\times C(48,2)}{C(52,5)}=\frac{4\times1128}{2598960}=\frac{4512}{2598960}\approx0.00174$.

Answer:

$0.00174$