An urn contains four balls bearing numbers 1, 2, 3 and 123 respectively. A ball is drawn at random from the urn. Let `E_i`, i = 1, 2, 3 denote the event that digit i appears on the ball drawn. Which one of the following is not true?

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

Answer

To solve the problem, we need to analyze the events \(E_1\), \(E_2\), and \(E_3\) based on the balls in the urn and determine which statement about their independence is not true.

### Step-by-Step Solution:

1. **Identify the Balls in the Urn:**
The urn contains four balls with the following numbers:
- Ball 1: 1
- Ball 2: 2
- Ball 3: 3
- Ball 4: 123

2. **Define Events:**
- \(E_1\): Event that digit 1 appears on the drawn ball.
- \(E_2\): Event that digit 2 appears on the drawn ball.
- \(E_3\): Event that digit 3 appears on the drawn ball.

3. **Calculate Probabilities of Each Event:**
- **For \(E_1\)**: The favorable cases are balls 1 and 4 (since both have the digit 1).
\[ P(E_1) = \frac{2}{4} = \frac{1}{2} \]
- **For \(E_2\)**: The favorable cases are balls 2 and 4 (since both have the digit 2).
\[ P(E_2) = \frac{2}{4} = \frac{1}{2} \]
- **For \(E_3\)**: The favorable cases are balls 3 and 4 (since both have the digit 3).
\[ P(E_3) = \frac{2}{4} = \frac{1}{2} \]

4. **Calculate Joint Probabilities:**
- **For \(E_1 \cap E_2\)**: The favorable case is only ball 4 (which has both digits 1 and 2).
\[ P(E_1 \cap E_2) = \frac{1}{4} \]
- **For \(E_2 \cap E_3\)**: The favorable case is only ball 4 (which has both digits 2 and 3).
\[ P(E_2 \cap E_3) = \frac{1}{4} \]
- **For \(E_1 \cap E_3\)**: The favorable case is only ball 4 (which has both digits 1 and 3).
\[ P(E_1 \cap E_3) = \frac{1}{4} \]
- **For \(E_1 \cap E_2 \cap E_3\)**: The favorable case is only ball 4 (which has digits 1, 2, and 3).
\[ P(E_1 \cap E_2 \cap E_3) = \frac{1}{4} \]

5. **Check Independence:**
- For events to be independent, the following must hold:
\[ P(E_i \cap E_j) = P(E_i) \cdot P(E_j) \]
- **Checking \(E_1\) and \(E_2\)**:
\[ P(E_1 \cap E_2) = \frac{1}{4} \quad \text{and} \quad P(E_1) \cdot P(E_2) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]
(They are independent.)
- **Checking \(E_2\) and \(E_3\)**:
\[ P(E_2 \cap E_3) = \frac{1}{4} \quad \text{and} \quad P(E_2) \cdot P(E_3) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]
(They are independent.)
- **Checking \(E_1\) and \(E_3\)**:
\[ P(E_1 \cap E_3) = \frac{1}{4} \quad \text{and} \quad P(E_1) \cdot P(E_3) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]
(They are independent.)
- **Checking \(E_1\), \(E_2\), and \(E_3\)**:
\[ P(E_1 \cap E_2 \cap E_3) = \frac{1}{4} \quad \text{and} \quad P(E_1) \cdot P(E_2) \cdot P(E_3) = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} \]
(They are **not independent**.)

6. **Conclusion:**
The statement that \(E_1\), \(E_2\), and \(E_3\) are independent is **not true**.

### Final Answer:
The statement that is not true is: **D. \(E_1\), \(E_2\), and \(E_3\) are independent.**