Suppose a sample space S consists of 4 elements, i.e., `S={alpha_1, alpha_2, alpha_3, alpha_4} " where " alpha_1, alpha_2, alpha_3, alpha_4` are pair wise mutually exclusive. Which function defines a probability function on S?
(1) `P(alpha_1)=1/2, P(alpha_2)=1/3, P(alpha_3)=1/4, P(alpha_4)=1/4`
(2) `P(alpha_1)=1/4, P(alpha_2)=-1/2, P(alpha_3)=7/9, P(alpha_4)=1/3`
(3) `P(alpha_1)=1/2, P(alpha_2)=1/4, P(alpha_3)=1/8, P(alpha_4)=1/8`
(4) `P(alpha_1)=1/2, P(alpha_2)=1/4, P(alpha_3)=1/8, P(alpha_4)=0`

Explanation

To determine which function defines a probability function on the sample space \( S = \{ \alpha_1, \alpha_2, \alpha_3, \alpha_4 \} \), we need to check the following criteria for each option:

1. The probabilities must be non-negative.
2. The sum of the probabilities must equal 1.

Let's evaluate each option step by step.

### Option 1:
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