One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent? (i) E: ‘the card drawn is a spade’ F: ‘the card drawn is an ace’ (ii) E: ‘the card drawn is black’ 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’
(i)
It is given that there are two events E and F , event E when the card drawn is spade and event F when the card drawn is an ace.
The total cards in a well shuffled deck are 52 and the numbers of spade are 13 and the numbers of ace are 4 .
The probability for event E is,
P ( E ) = 13 52 = 1 4
The probability for event F is,
P ( F ) = 4 52 = 1 13
The intersection of the events is E ∩ F such that the card drawn is spade and an ace.
Since the number of ace of spades is 1 .
The probability can be written as,
P ( E ∩ F ) = 1 52 (1)
Condition for independent events is,
P ( E ) ⋅ P ( F ) = 1 4 × 1 13 = 1 52 (2)
From equation (1) and (2),
P ( E ∩ F ) = P ( E ) ⋅ P ( F )
Hence, the events E and F are independent events.
(ii)
It is given that there are two events E and F , event E when the card drawn is black and event F when the card drawn is a king.
The total cards in a well shuffled deck are 52 and the numbers of black cards are 26 and the numbers of king are 4 .
The probability for event E is,
P ( E ) = 26 52 = 1 2
The probability for event F is,
P ( F ) = 4 52 = 1 13
The intersection of the events is E ∩ F such that the card drawn is black and a king.
Since, the number of black king is 2 .
The probability can be written as,
P ( E ∩ F ) = 2 52 = 1 26 (1)
Condition for independent events is,
P ( E ) ⋅ P ( F ) = 1 2 × 1 13 = 1 26 (2)
From equation (1) and (2),
P ( E ∩ F ) = P ( E ) ⋅ P ( F )
Hence, the events E and F are independent events.
(iii)
It is given that there are two events E and F , event E when the card drawn is king or queen and event F when the card drawn is a queen or jack.
The total cards in a well shuffled deck are 52 and the numbers of kings or queen are 8 and the numbers of queen or jack are 4 .
The probability for event E is,
P ( E ) = 8 52 = 2 13
The probability for event F is,
P ( F ) = 8 52 = 2 13
The intersection of the events is E ∩ F such that the card drawn is a king or queen and queen or jack.
Since, the number of queens are 4 .
The probability can be written as,
P ( E ∩ F ) = 4 52 = 1 13 (1)
Condition for independent events is,
P ( E ) ⋅ P ( F ) = 2 13 × 2 13 = 4 169 (2)
From the equation (1) and (2),
P ( E ∩ F ) ≠ P ( E ) ⋅ P ( F )
Hence, the events E and F are not the independent events.
(i)
It is given that there are two events
The total cards in a well shuffled deck are
The probability for event E is,
The probability for event F is,
The intersection of the events is
Since the number of ace of spades is
The probability can be written as,
Condition for independent events is,
From equation (1) and (2),
Hence, the events
(ii)
It is given that there are two events
The total cards in a well shuffled deck are
The probability for event E is,
The probability for event F is,
The intersection of the events is
Since, the number of black king is
The probability can be written as,
Condition for independent events is,
From equation (1) and (2),
Hence, the events
(iii)
It is given that there are two events
The total cards in a well shuffled deck are
The probability for event E is,
The probability for event F is,
The intersection of the events is
Since, the number of queens are
The probability can be written as,
Condition for independent events is,
From the equation (1) and (2),
Hence, the events
