a standard deck of 52 playing cards contains 13 cards in each of four suits: hearts, diamonds, clubs, and…

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%

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%

Answer

Answer:

  1. Discipline and Sub - field:
    • Discipline: Mathematics
    • Sub - field: Statistics
  2. 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%.