One card is drawn at random from a pack of well-shuffled deck of 52 cards. Check whether the following events are independent :
E: 'the card drawn is black'
F: 'the card drawn is a king'.

Answer

To determine whether the events E (the card drawn is black) and F (the card drawn is a king) are independent, we need to follow these steps:

### Step 1: Define the Events
- Let E be the event that the card drawn is black.
- Let F be the event that the card drawn is a king.

### Step 2: Calculate the Probability of Event E
In a standard deck of 52 cards, there are 26 black cards (13 spades and 13 clubs).
- Therefore, the probability of event E, \( P(E) \), is given by:
\[ P(E) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \]

### Step 3: Calculate the Probability of Event F
In a standard deck, there are 4 kings (1 king of hearts, 1 king of diamonds, 1 king of clubs, and 1 king of spades).
- Therefore, the probability of event F, \( P(F) \), is given by:
\[ P(F) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \]

### Step 4: Calculate the Probability of the Intersection of Events E and F
The intersection of events E and F, \( P(E \cap F) \), is the probability that the card drawn is both black and a king. There are 2 black kings (king of spades and king of clubs).
- Therefore, the probability of the intersection is:
\[ P(E \cap F) = \frac{\text{Number of black kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26} \]

### Step 5: Check for Independence
Two events E and F are independent if:
\[ P(E \cap F) = P(E) \times P(F) \]
Calculating the right side:
\[ P(E) \times P(F) = \left(\frac{1}{2}\right) \times \left(\frac{1}{13}\right) = \frac{1}{26} \]

### Step 6: Conclusion
Since \( P(E \cap F) = \frac{1}{26} \) and \( P(E) \times P(F) = \frac{1}{26} \), we conclude that:
\[ P(E \cap F) = P(E) \times P(F) \]
Thus, events E and F are independent.

### Final Answer
Yes, the events E and F are independent.

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