One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events and independent?
(i) E: ‘the card drawn is a spade’ F: ‘the card drawn is an ace’
(ii) E: ‘the card drawn is black’ : ‘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 and independent?
(i) E: ‘the card drawn is a spade’ F: ‘the card drawn is an ace’
(ii) E: ‘the card drawn is black’ : ‘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, the card drawn is a spade

the card drawn is an ace

the card drawn is a spade and ace both (1)

And

From (1) and (2)

Hence, and 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, the card drawn is of black

the card drawn is a king

the card drawn is a black and king both (1)

And

From (1) and (2)

Hence, and are independent events.

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

Hence, the card drawn is either king or queen

the card drawn is either queen or jack

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

the card draw is either king or queen and either queen or jack (1)

And

From (1) and (2)

Hence, and are not independent events.