A bag contains four tickets marked with numbers `112,121,211,` and `222`. One ticket is drawn at random from the bag. Let `E_(i)(i=1,2,3)` denote the event that `ith` digit on the ticket is `2`. Then

Options

A. `E_(1)` and `E_(2)` are independent
B. `E_(2)` and `E_(3)` are independent
C. `E_(3)` and `E_(1)` are independent
D. `E_(1)`,`E_(2)`.`E_(3)` are independent

Explanation

To solve the problem, we need to analyze the events \( E_1, E_2, \) and \( E_3 \) based on the tickets drawn from the bag.

### Step 1: Define the Events
Let’s define the events based on the digits of the tickets:
- \( E_1 \): The event that the first digit of the ticket is 2.
- \( E_2 \): The event that the second digit of the ticket is 2.
- \( E_3 \): The event that the third digit of the ticket is 2.

### Step 2: List the Tickets
The tickets in the bag are:
1. 112
2. 121
3. 211
4. 222

### Step 3: Determine the Outcomes for Each Event
Now we will determine which tickets correspond to each event:
- **For \( E_1 \)** (First digit is 2):
- Tickets: None (0 tickets)

- **For \( E_2 \)** (Second digit is 2):
- Tickets: 112, 222 (2 tickets)

- **For \( E_3 \)** (Third digit is 2):
- Tickets: 112, 121, 222 (3 tickets)

### Step 4: Calculate Probabilities
Now we calculate the probabilities of each event:
- \( P(E_1) = \frac{0}{4} = 0 \)
- \( P(E_2) = \frac{2}{4} = \frac{1}{2} \)
- \( P(E_3) = \frac{3}{4} \)

### Step 5: Check Independence of Events
Two events \( E_i \) and \( E_j \) are independent if:
\[ P(E_i \cap E_j) = P(E_i) \cdot P(E_j) \]

1. **Check \( E_1 \) and \( E_2 \)**:
- \( P(E_1 \cap E_2) = 0 \)
- \( P(E_1) \cdot P(E_2) = 0 \cdot \frac{1}{2} = 0 \)
- **Independent**.

2. **Check \( E_2 \) and \( E_3 \)**:
- \( P(E_2 \cap E_3) = P(\{112, 222\} \cap \{112, 121, 222\}) = P(\{112, 222\}) = \frac{2}{4} = \frac{1}{2} \)
- \( P(E_2) \cdot P(E_3) = \frac{1}{2} \cdot \frac{3}{4} = \frac{3}{8} \)
- **Not Independent**.

3. **Check \( E_1 \) and \( E_3 \)**:
- \( P(E_1 \cap E_3) = 0 \)
- \( P(E_1) \cdot P(E_3) = 0 \cdot \frac{3}{4} = 0 \)
- **Independent**.

### Step 6: Conclusion
From the checks above, we conclude:
- \( E_1 \) and \( E_2 \) are independent.
- \( E_2 \) and \( E_3 \) are not independent.
- \( E_1 \) and \( E_3 \) are independent.

Therefore, the correct options are that \( E_1 \) and \( E_2 \) are independent, \( E_1 \) and \( E_3 \) are independent, but \( E_2 \) and \( E_3 \) are not independent.