1.

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

Answer»

Given: Pack of 52 cards.

By using the formula,

P (E) = favourable outcomes / total possible outcomes

We know that, a card is drawn from a pack of 52 cards, so 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)

P (E) = n (E) / n (S)

= 2 / 52

= 1/26

(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.]

P (E) = n (E) / n (S)

= 28 / 52

= 7/13

(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)

P (E) = n (E) / n (S)

= 2 / 52

= 1/26

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

n (E) = 4C1 + 4C1 + 4C1 = 12

P (E) = n (E) / n (S)

= 12 / 52

= 3/13

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

Now let us 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′) = n (E′) / n (S)

= 16 / 52

= 4/13

So, P (E) = 1 – P (E′)

= 1 – 4/13

= 9/13

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

n (E) = 13C1 + 4C1 - 1 = 16

P (E) = n (E) / n (S)

= 16 / 52

= 4/13

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

Now let us consider E′ as the event that either an ace or king appears

n(E′) = 4C1 + 4C1 = 8

P (E′) = n (E′) / n (S)

= 8 / 52

= 2/13

So, P (E) = 1 – P (E′)

= 1 – 2/13

= 11/13

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

n (E)=13C1=13

P (E) = n (E) / n (S)

= 13 / 52

= 1/4

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

Now let us consider E′ as the event that diamond card appears

n (E′) =13C1=13

P (E′) = n (E′) / n (S)

= 13 / 52

= 1/4

So, P (E) = 1 – P (E′)

= 1 – 1/4

= 3/4

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

n (E) =26C1 = 26 (spades and clubs)

P (E) = n (E) / n (S)

= 26 / 52

= 1/2

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

Now let us consider E′ as the event that ace card appears

n (E′) = 4C1 = 4

P (E′) = n (E′) / n (S)

= 4 / 52

= 1/13

So, P (E) = 1 – P (E′)

= 1 – 1/13

=12/13

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

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

P (E) = n (E) / n (S)

= 26 / 52

= 1/2


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