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)