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 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

Answer 1

A pack of cards have 52 cards, 26 black and four kinds each of 13 cards from 2 to 10, one ace, one ace, one jack, one queen and one king.

$\therefore$ Total number of possible events = 52

(i) Let A be the occurence of favourable events which is a black king which are 2.

$\therefore P\left(A\right)=\frac{2}{52}=\frac{1}{26}$

(ii) Let B be the occurence of favourable events such that it is either a black card or a king.

Total = number of black cards = 26 + 2 red kings = 28

$\therefore P\left(B\right)=\frac{28}{52}=\frac{7}{13}$

(iii) Let C be the occurence of favourable events such that it is black and a king which can be 2.

$\therefore P\left(C\right)=\frac{2}{52}=\frac{1}{26}$

(iv) Let D be the occurence of favourable events such that it is a jack, queen or a king which will be 4 + 4 + 4 = 12

$\therefore P\left(D\right)=\frac{12}{52}=\frac{3}{13}$

(v) Let E be the occurence of favourable events such that it is neither a heart nor a king.

$\therefore$ Number of favourable event will be

$13\times 3-3=39-3=36$

$\therefore P\left(E\right)=\frac{36}{52}=\frac{9}{13}$

(vi) Let F be the occurence of favourable events such that it is a spade or an ace.

$\therefore$ Number of events = 13 + 3 = 16

$\therefore P\left(F\right)=\frac{16}{52}=\frac{4}{13}$

(vii) Let G be the occurence of favourable events such that it neither an ace nor a king.

$\therefore$ Number of events = 52 - 4 - 4 = 4

$\therefore P\left(G\right)=\frac{44}{52}=\frac{11}{13}$

(viii) Let H be the occurence of favourable events such that it is neither a red card nor a queen.

$\therefore$ Number of events = 26 - 2 = 24

$\therefore P\left(H\right)=\frac{24}{52}=\frac{6}{13}$

(ix) Let I be the occurence of favourable events such that it is other than an ace.

$\therefore$ Number of events = 52 - 4 = 48

$\therefore P\left(I\right)=\frac{48}{52}=\frac{12}{13}$

(x) Let J be the occurence of favourable event such that it is ten

$\therefore$ Number of events = 4

$\therefore P\left(J\right)=\frac{4}{52}=\frac{1}{13}$

(xi) Let K be the occurence of favourable event such that it is a spade.

$\therefore$ Number of events = 13

$\therefore P\left(K\right)=\frac{13}{52}=\frac{1}{4}$

(xii) Let L be the occurence of favourable event such that it is a black card.

$\therefore$ Number events = 26

$\therefore P\left(L\right)=\frac{26}{52}=\frac{1}{2}$

(xiii) Let M be the occurence of favourable event such that it is the seven of clubs.

$\therefore$ Number of events = 1

$\therefore P\left(M\right)=\frac{1}{52}$

(xiv) Let N be the occurence of favourable event such that it is a jack.

$\therefore$ Number of events = 4

$\therefore P\left(N\right)=\frac{4}{52}=\frac{1}{13}$

(xv) Let O be the occurence of favourable event such that it is an ace of spades.

$\therefore$ Number of events = 1

$\therefore P\left(O\right)=\frac{1}{52}$

(xvi) Let Q be the occurence of favourable event such that it is a queen.

$\therefore$ Number of events = 4

$\therefore P\left(Q\right)=\frac{4}{52}=\frac{1}{13}$

(xvii) Let R be the occurence of favourable event such that it is a heart card.

$\therefore$ Number of events = 13

$\therefore P\left(R\right)=\frac{13}{52}=\frac{1}{4}$

(xviii) Let S be the occurence of favourable event such that it is a red card

$\therefore$ Number of events = 26

$\therefore P\left(S\right)=\frac{26}{52}=\frac{1}{2}$