A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is ;

(i) a black king


(ii) either a black card or a king


(iii) black and a king


(iv) a jack, queen or a king


(v) neither an ace nor a king


(vi) spade or an ace


(vii) neither an ace nor a king


(viii) a diamond card


(ix) not a diamond card


(x) a black card


(xi) not an ace


(xii) not a black card

given: pack of 52 cards

formula:


since a card is drawn from a pack of 52 cards, therefore number of elementary events in the sample space is


n(S)= 52C1 = 52


(i) let E be the event of drawing a black king


n(E)=2C1 =2 (there are two black kings one of spade and other of club)




(ii) let E be the event of drawing a black card or a king


n(E)=26C1+4C1-2C1=28


we are subtracting 2 from total because there are two black king which are already counted and to avoid the error of considering it twice




(iii) let E be the event of drawing a black card and a king


n(E)=2C1 =2 (there are two black kings one of spade and other of club)




(iv) let E be the event of drawing a jack, queen or king


n(E)=4C1+4C1+4C1=12




(v) let E be the event of drawing neither a heart nor a king


now consider E’ as the event that either a heart or king appears


n(E’) =6C1+4C1-1=16 (there is a heart king so it is deducted)




P(E)=1-P(E’)



(vi) let E be the event of drawing a spade or king


n(E)=13C1+4C1-1=16




(vii) let E be the event of drawing neither an ace nor a king


now consider E’ as the event that either an ace or king appears


n(E’) =4C1+4C1=8




P(E)=1-P(E’)



(viii) let E be the event of drawing a diamond card


n(E)=13C1=13




(ix) let E be the event of drawing not a diamond card


now consider E’ as the event that diamond card appears


n(E’) =13C1=13




P(E)=1-P(E’)



(x) let E be the event of drawing a black card


n(E)=26C1=26 (spades and clubs)




(xi) let E be the event of drawing not an ace


now consider E’ as the event that ace card appears


n(E’) =4C1=4




P(E)=1-P(E’)



(xii) let E be the event of not drawing a black card


n(E)=26C1=26 (red cards of hearts and diamonds)




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