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.