A card is drawn from a well shuffled pack of 52 cards . In which of the following cases, the events E and F are independent events?
(i) `E=` card drawn is heart.
`F=` card drawn is a king.
(ii) `E=` card drawn is red.
`F=` card drawn is a king.

Explanation

To determine whether the events E and F are independent in the given scenarios, we need to check if the probability of the intersection of E and F equals the product of their individual probabilities. Let's analyze both cases step by step.

### Case (i):
- **Event E**: Card drawn is a heart.
- **Event F**: Card drawn is a king.

**Step 1: Calculate P(E)**
- There are 13 hearts in a deck of 52 cards.
- Therefore, the probability of drawing a heart (P(E)) is:
\[ P(E) = \frac{13}{52} = \frac{1}{4} \]

**Step 2: Calculate P(F)**
- There are 4 kings in a deck of 52 cards.
- Therefore, the probability of drawing a king (P(F)) is:
\[ P(F) = \frac{4}{52} = \frac{1}{13} \]

**Step 3: Calculate P(E ∩ F)**
- The intersection of E and F (E ∩ F) is the event that the card drawn is both a heart and a king. There is only one king of hearts.
- Therefore, the probability of E intersection F (P(E ∩ F)) is:
\[ P(E ∩ F) = \frac{1}{52} \]

**Step 4: Check for independence**
- For E and F to be independent, the following must hold:
\[ P(E ∩ F) = P(E) \times P(F) \]
- Calculating the right side:
\[ P(E) \times P(F) = \frac{1}{4} \times \frac{1}{13} = \frac{1}{52} \]
- Since \( P(E ∩ F) = P(E) \times P(F) \), events E and F are independent.

### Case (ii):
- **Event E**: Card drawn is red.
- **Event F**: Card drawn is a king.

**Step 1: Calculate P(E)**
- There are 26 red cards (13 hearts and 13 diamonds) in a deck of 52 cards.
- Therefore, the probability of drawing a red card (P(E)) is:
\[ P(E) = \frac{26}{52} = \frac{1}{2} \]

**Step 2: Calculate P(F)**
- As calculated before, the probability of drawing a king (P(F)) is:
\[ P(F) = \frac{4}{52} = \frac{1}{13} \]

**Step 3: Calculate P(E ∩ F)**
- The intersection of E and F (E ∩ F) is the event that the card drawn is both red and a king. There are 2 red kings (king of hearts and king of diamonds).
- Therefore, the probability of E intersection F (P(E ∩ F)) is:
\[ P(E ∩ F) = \frac{2}{52} = \frac{1}{26} \]

**Step 4: Check for independence**
- For E and F to be independent:
\[ P(E ∩ F) = P(E) \times P(F) \]
- Calculating the right side:
\[ P(E) \times P(F) = \frac{1}{2} \times \frac{1}{13} = \frac{1}{26} \]
- Since \( P(E ∩ F) = P(E) \times P(F) \), events E and F are independent.

### Conclusion:
In both cases (i) and (ii), the events E and F are independent.