Question

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 Select the correct answer from above options

Answer 1

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