A sample space consists of 4 elements `S=(alpha_(1), alpha_(2), alpha_(3), alpha_(4))` Which defines a probability function on S `(where" "alpha_(i)capalpha_(j)nephi" "AA" "i nejand cupalpha_(i)=S)`

Options

A. `P(alpha_(1))=1/2,P(alpha_(2))=1/3,P(alpha_(3))=1/4,P(alpha_(4))=1/5`
B. `P(alpha_(1))=1/2,P(alpha_(2))=1/4,P(alpha_(3))=-1/4,P(alpha_(4))=1/2`
C. `P(alpha_(1))=1/2,P(alpha_(2))=1/4,P(alpha_(3))=1/8,P(alpha_(4))=1/8`
D. `P(alpha_(1))=1/2,P(alpha_(2))=1/4,P(alpha_(3))=1/8,P(alpha_(4))=0`

Answer

To solve the problem, we need to analyze the probability functions defined on the sample space \( S = \{ \alpha_1, \alpha_2, \alpha_3, \alpha_4 \} \). We will check each case to determine which one satisfies the properties of a probability function.

### Step-by-Step Solution:

1. **Understanding Probability Function Properties**:
- The probability of any event must be between 0 and 1, inclusive.
- The sum of the probabilities of all outcomes in the sample space must equal 1.

2. **Evaluate Each Case**:
- **Case 1**: \( P(\alpha_1) = \frac{1}{2}, P(\alpha_2) = \frac{1}{3}, P(\alpha_3) = \frac{1}{4}, P(\alpha_4) = \frac{1}{5} \)
- Calculate the total probability:
\[ P(\alpha_1) + P(\alpha_2) + P(\alpha_3) + P(\alpha_4) = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \]
- Finding a common denominator (60):
\[ = \frac{30}{60} + \frac{20}{60} + \frac{15}{60} + \frac{12}{60} = \frac{77}{60} > 1 \]
- **Conclusion**: This case does not define a valid probability function.

- **Case 2**: \( P(\alpha_1) = \frac{1}{4}, P(\alpha_2) = -\frac{1}{4}, P(\alpha_3) = \frac{1}{4}, P(\alpha_4) = \frac{1}{4} \)
- Here, \( P(\alpha_2) < 0 \), which violates the non-negativity condition of probabilities.
- **Conclusion**: This case does not define a valid probability function.

- **Case 3**: \( P(\alpha_1) = \frac{1}{2}, P(\alpha_2) = \frac{1}{4}, P(\alpha_3) = \frac{1}{8}, P(\alpha_4) = \frac{1}{8} \)
- Calculate the total probability:
\[ P(\alpha_1) + P(\alpha_2) + P(\alpha_3) + P(\alpha_4) = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{8} \]
- Finding a common denominator (8):
\[ = \frac{4}{8} + \frac{2}{8} + \frac{1}{8} + \frac{1}{8} = \frac{8}{8} = 1 \]
- **Conclusion**: This case defines a valid probability function.

- **Case 4**: \( P(\alpha_1) = \frac{1}{2}, P(\alpha_2) = \frac{1}{4}, P(\alpha_3) = \frac{1}{8}, P(\alpha_4) = 0 \)
- Calculate the total probability:
\[ P(\alpha_1) + P(\alpha_2) + P(\alpha_3) + P(\alpha_4) = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + 0 \]
- Finding a common denominator (8):
\[ = \frac{4}{8} + \frac{2}{8} + \frac{1}{8} + 0 = \frac{7}{8} < 1 \]
- **Conclusion**: This case does not define a valid probability function.

3. **Final Conclusion**:
- The only case that satisfies the properties of a probability function is **Case 3**.