A card is drawn at random form a well-shuffled deck of playing cards.
Question:

A card is drawn at random form a well-shuffled deck of playing cards. Find the probability that the card drawn is
(i) a card of spades of an ace
(ii) a red king
(iii) either a king or a queen
(iv) neither a king nor a queen.

Solution:

Total number of all possible outcomes= 52
(i) Number of spade cards = 13
Number of aces = 4 (including 1 of spade)

Therefore, number of spade cards and aces = (13 + 4 − 1) = 16  
$\therefore P($ getting a spade or an ace card $)=\frac{16}{52}=\frac{4}{13}$
 
(ii) Number of  red kings = 2
$\therefore P($ getting a red king $)=\frac{2}{52}=\frac{1}{26}$
 
(iii) Total number of kings = 4
     Total number of queens = 4
      Let E be the event of getting either a king or a queen.
      Then, the favourable outcomes =  4 + 4 = 8 
$\therefore P($ getting a king or a queen $)=P(E)=\frac{8}{52}=\frac{2}{13}$
 
(iv)  Let E be the event of getting either a king or a queen. Then, ( not E) is the event that drawn card is neither a king nor a
       queen.  
 
Then, $P$ (getting a king or a queen ) $=\frac{2}{13}$
 
  Now, ​P (E) + ​P (not E) = 1
 
$\therefore P($ getting neither a king nor a queen $)=1-\left(\frac{2}{13}\right)=\frac{11}{13}$
 
 
 
 
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