One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events \mathrm{E} and \mathrm{F} independent?
(i) E: ‘the card drawn is a spade’ F: ‘the card drawn is an ace’
(ii) E: ‘the card drawn is black’ \mathrm{F} : ‘the card drawn is a king’
(iii) E: ‘the card drawn is a king or queen’ F: ‘the card drawn is a queen or jack’.
One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events \mathrm{E} and \mathrm{F} independent?
(i) E: ‘the card drawn is a spade’ F: ‘the card drawn is an ace’
(ii) E: ‘the card drawn is black’ \mathrm{F} : ‘the card drawn is a king’
(iii) E: ‘the card drawn is a king or queen’ F: ‘the card drawn is a queen or jack’.

Solution:

Given: A deck of 52 cards.

Concept: Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.
Calculation:

(i) In a deck of 52 cards, there are 13 spades, 4 aces, and only one card that is both a spade and an ace.

Hence, P(E)= the card drawn is a spade =13 / 52=1 / 4

\mathrm{P}(\mathrm{F})= the card drawn is an ace =4 / 52=1 / 13

P(E \cap F)= the card drawn is a spade and ace both =1 / 52 \ldots . . (1)

And P(E) . P(F)

=1 / 4 \times 1 / 13=1 / 52 \ldots .(2)

From (1) and (2)

\Rightarrow P(E \cap F)=P(E) . P(F)

Hence, \mathrm{E} and \mathrm{F} are independent events.

(ii) In a deck of 52 cards, 26 cards are black, 4 cards are king, and there are only two cards that are both black and king at the same time.

Hence, P(E)= the card drawn is of black =26 / 52=1 / 2

P(F)= the card drawn is a king =4 / 52=1 / 13

P(E \cap F)= the card drawn is a black and king both =2 / 52=1 / 26 \ldots (1)

And P(E) . P(F)

=1 / 2 \times 1 / 13=1 / 26 \ldots .(2)

From (1) and (2)

\Rightarrow P(E \cap F)=P(E) \cdot P(F)

Hence, \mathrm{E} and \mathrm{F} are independent events.

(iii) In a standard 52-card deck, the queen, the king, and the jack are represented by four cards each.

Hence, \mathrm{P}(\mathrm{E})= the card drawn is either king or queen =8 / 52=2 / 13

\mathrm{P}(\mathrm{F})= the card drawn is either queen or jack =8 / 52=2 / 13

There are 4 cards which are either king or queen and either queen or jack.

P(E \cap F)= the card draw n is either king or queen and either queen or jack =4 / 52=1 / 13 \ldots (1)

And P(E) . P(F)

=2 / 13 \times 2 / 13=4 / 169 . \ldots .

From (1) and (2)

\Rightarrow P(E \cap F) \neq P(E) . P(F)

Hence, \mathrm{E} and \mathrm{F} are not independent events.