A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is:
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 a heart nor a king

(vi) spade or an ace

(vii) neither an ace nor a king

(viii) neither a red card nor a queen.

(ix) other than an ace

(x) a ten

(xi) a spade

(xii) a black card

(xiii) the seven of clubs

(xiv) jack

(xv) the ace of spades

(xvi) a queen

(xvii) a heart

(xviii) a red card

Solution:

(i) There are two black kings, spade and clover. Hence, the probability that the drawn card is a black king is: 2/52 = 1/26

(ii) There are 26 black cards and 4 kings, but two kings are already black. Hence, we only need to count the red kings. Thus, the probability is: (26+2)/52 = 7/13

(iii) This question is exactly the same as part (i). Hence, the probability is: 2/52 = 1/26

(iv) There are 4 jacks, 4 queens and 4 kings in a deck. Hence, the probability of drawing either of them is: (4+4+4)/52 = 3/13

(v) This means that we have to leave the hearts and the kings out. There are 13 hearts and 3 kings (other than that of hearts). Hence, the probability of drawing neither a heart nor a king is: (52-13-3)/52 = 9/13

(vi) There are 13 spades and 3 aces (other than that of spades). Hence the probability is: (13+3)/52 = 4/13

(vii) This means that we have to leave the aces and the kings out. There are 4 aces and 4 kings. Hence, the probability of drawing neither an ace nor a king is: $(52-4-4) / 52=11 / 13$.

(viii) This means that we have to leave the red cards and the queens out. There are 26 red cards and 2 queens (only black queens are counted since the reds are already counted among the red cards). Hence, the probability of drawing neither a red card nor a queen is: (52-26-2)/52 = 6/13

(ix) It means that we have to leave out the aces. Since there are 4 aces, then the probability is (524)/52 = 12/13

(x) Since there are four 10s, the probability is: 4/52 = 1/13

(xi) Since there are 13 spades, the probability is: 13/52 = 1/4

(xii) Since there are 26 black cards, the probability is: 26/52 = 1/2

(xiii) There is only one card named seven of clubs. Hence, the probability is 1/52.

(xiv) Since there are 4 jacks, the probability is: 4/52 = 1/13

(xv) There is only 1 card named ace of spade. Hence, the probability is 1/52.

(xvi) Since there are 4 queens, the probability is: 4/52 = 1/13

(xvii) Since there are 13 hearts, the probability is: 13/52 = 1/4

(xviii) Since there are 26 red cards, the probability is 26/52 = 1/2

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