Which of the following cannot be valid assignmentof probabilities for outcomes of sample space
`S={omega_(1),omega_(2),omega_(3),omega_(4),omega_(5),omega_(6),omega_(7)}`?
Assignement `omega_(1),omega_(2),omega_(3),omega_(4),omega_(5),omega_(6),omega^(7)`
a. 0.1,0.01,0.05,0.03,0.01,0.3,0.6
b. `1/7,1/7,1/7,1/7,1/7,1/7,1/7`
c. 0.1,0.2,0.3,0.4,0.5,0.6,0.7
d. -0.1,0.2,0.3,0.4,-0.2,0.1,0.3
e. `1/14,2/14,3/14,4/14,5/14,6/14,15/14`

Answer

To determine which of the given assignments of probabilities cannot be valid for the outcomes of the sample space \( S = \{ \omega_1, \omega_2, \omega_3, \omega_4, \omega_5, \omega_6, \omega_7 \} \), we need to check two main conditions for each assignment:

1. Each probability must be between 0 and 1 (inclusive).
2. The sum of all probabilities must equal 1.

Let's analyze each assignment step by step.

### Step 1: Analyze Assignment (a)
Probabilities: \( 0.1, 0.01, 0.05, 0.03, 0.01, 0.3, 0.6 \)

**Check individual probabilities:**
- \( 0.1 \) is valid
- \( 0.01 \) is valid
- \( 0.05 \) is valid
- \( 0.03 \) is valid
- \( 0.01 \) is valid
- \( 0.3 \) is valid
- \( 0.6 \) is valid

**Sum of probabilities:**
\[ 0.1 + 0.01 + 0.05 + 0.03 + 0.01 + 0.3 + 0.6 = 1.1 \]
Since the sum is greater than 1, this assignment is **not valid**.

### Step 2: Analyze Assignment (b)
Probabilities: \( \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7}, \frac{1}{7} \)

**Check individual probabilities:**
- Each \( \frac{1}{7} \) is valid.

**Sum of probabilities:**
\[ 7 \times \frac{1}{7} = 1 \]
This assignment is **valid**.

### Step 3: Analyze Assignment (c)
Probabilities: \( 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 \)

**Check individual probabilities:**
- All values are valid.

**Sum of probabilities:**
\[ 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 = 2.8 \]
Since the sum is greater than 1, this assignment is **not valid**.

### Step 4: Analyze Assignment (d)
Probabilities: \( -0.1, 0.2, 0.3, 0.4, -0.2, 0.1, 0.3 \)

**Check individual probabilities:**
- \( -0.1 \) is not valid.
- \( -0.2 \) is not valid.

Since there are negative probabilities, this assignment is **not valid**.

### Step 5: Analyze Assignment (e)
Probabilities: \( \frac{1}{14}, \frac{2}{14}, \frac{3}{14}, \frac{4}{14}, \frac{5}{14}, \frac{6}{14}, \frac{15}{14} \)

**Check individual probabilities:**
- \( \frac{1}{14}, \frac{2}{14}, \frac{3}{14}, \frac{4}{14}, \frac{5}{14}, \frac{6}{14} \) are valid.
- \( \frac{15}{14} \) is not valid (greater than 1).

Since one of the probabilities is greater than 1, this assignment is **not valid**.

### Conclusion
The assignments that are **not valid** are:
- (a) \( 0.1, 0.01, 0.05, 0.03, 0.01, 0.3, 0.6 \)
- (c) \( 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 \)
- (d) \( -0.1, 0.2, 0.3, 0.4, -0.2, 0.1, 0.3 \)
- (e) \( \frac{1}{14}, \frac{2}{14}, \frac{3}{14}, \frac{4}{14}, \frac{5}{14}, \frac{6}{14}, \frac{15}{14} \)