Question

a standard deck of 52 playing cards contains 13 cards in each of four suits: hearts, diamonds, clubs, and spades. four cards are drawn from the deck at random. what is the approximate probability that exactly three of the cards are diamonds? 1% 4% 44%

Explanation:

Answer:

1. Discipline and Sub - field:

• Discipline: Mathematics
• Sub - field: Statistics

1. Answer:

• First, use the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\). The total number of ways to draw 4 cards from 52 cards is \(C(52,4)=\frac{52!}{4!(52 - 4)!}=\frac{52\times51\times50\times49}{4\times3\times2\times1}=270725\).
• The number of ways to choose 3 diamonds out of 13 is \(C(13,3)=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286\).
• The number of ways to choose 1 non - diamond card out of \(52 - 13=39\) non - diamond cards is \(C(39,1)=\frac{39!}{1!(39 - 1)!}=39\).
• The number of ways to draw exactly 3 diamonds and 1 non - diamond card is \(C(13,3)\times C(39,1)=286\times39 = 11154\).
• The probability \(P=\frac{C(13,3)\times C(39,1)}{C(52,4)}=\frac{11154}{270725}\approx0.0412\approx4\%\)
• So the answer is 4%.