Question

One card is drawn at random from a well shuffled pack of 52 cards. In which of the following cases are the events E and F independent?

1. E: ‘the card drawn is a spades.’
F: ‘the card drawn is an ace.’

2. E: ‘the card drawn is a black.’
F: ‘the card drawn is a king.’

3. E: ‘the card drawn is a king or a queen.’
F: ‘the card drawn is queen or a jack.’

Answer 1

1. P(E) = \(\frac{13}{52}\) =\(\frac{1}{4}\), P(F) = \(\frac{4}{52}\) = \(\frac{1}{13}\)

There is only one card which is an ace of spade.

P(E ∩ F) = \(\frac{1}{52}\)

We have,

P(E) × P(F) = \(\frac{1}{4} \times \frac{1}{13} = \frac{1}{52}\) = P(E ∩ F)

Hence E and F are independent events.

2. P(E) = \(\frac{26}{52}\) = \(\frac{1}{2}\), P(F) = \(\frac{4}{52} = \frac{1}{13}\)

There are two king of black.

P(E ∩ F) = \(\frac{2}{52} = \frac{1}{26}\)

We have,

P(E) × P(F) = \(\frac{1}{2} \times \frac{1}{13} = \frac{1}{26}\)= P(E ∩ F)

Hence E and F are independent events.

3. There are 4 king and 4 queen cards

P(E) = \(\frac{8}{52} = \frac{2}{13}\)

There are 4 queen and 4 jack cards.

P(F) = \(\frac{8}{52} = \frac{2}{13}\)

There 4 queen common for both.

P(E ∩ F) = \(\frac{4}{52} = \frac{1}{13}\)

We have,

P(E) × P(F) = \(\frac{2}{13} \times \frac{2}{13} = \frac{4}{169} \neq\)  P(E ∩ F)

Hence E and F are not independent events.