Question

if you are dealt 3 cards from a shuffled deck of 52 cards, find the probability that all 3 cards are clubs. click the icon to view a description of a standard deck of playing cards. the probability is \boxed{}. (round to six decimal places as needed.)

Explanation:

Step1: Determine number of clubs and total cards

A standard deck has 52 cards, with 13 clubs. We want to find the probability of choosing 3 clubs from 13, and 3 cards from 52. The combination formula is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n \) is the total number, \( k \) is the number chosen.

Step2: Calculate combinations for clubs and total

For clubs: \( C(13, 3)=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286 \)
For total: \( C(52, 3)=\frac{52!}{3!(52 - 3)!}=\frac{52\times51\times50}{3\times2\times1}=22100 \)

Step3: Find the probability

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

Answer:

\( 0.012941 \)