Question

if you are dealt 3 cards from a shuffled deck of 52 cards, find the probability that all 3 cards are diamonds
click the icon to view a description of a standard deck of playing cards

the probability is
(round to six decimal places as needed.)

Explanation:

Step1: Calculate the number of ways to choose 3 cards from 52 cards

The combination formula is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 52\) and \(r=3\).

$$C(52,3)=\frac{52!}{3!(52 - 3)!}=\frac{52\times51\times50}{3\times2\times1}=22100$$

Step2: Calculate the number of ways to choose 3 diamond cards from 13 diamond cards

Using the combination formula with \(n = 13\) and \(r = 3\)

$$C(13,3)=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286$$

Step3: Calculate the probability

The probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)

$$P=\frac{C(13,3)}{C(52,3)}=\frac{286}{22100}\approx0.012941$$

Answer:

\(0.012941\)