- A probability distribution must satisfy two conditions: all individual probabilities must be non-negative (\(P(x_i) \ge 0\)) and their total sum must equal \(1\) (\(\sum P(x_i) = 1\)).
- Table 1, Table 3, and Table 4 fail the non-negativity test because they contain negative values.
- Table 2 contains all positive values, and their sum is \(0.11 + 0.05 + 0.49 + 0.35 = 1.0\).
- \(\boxed{\text{Table 2}}\)
Explanation
- Analyze the conditions for a probability distribution
A probability distribution for a discrete random variable \(X\) is defined by two fundamental conditions that must both be satisfied:
- Non-negativity: Each individual probability must be greater than or equal to zero, i.e., \(P(x_i) \geq 0\) for every outcome \(x_i\).
- Normalization: The sum of all probabilities for all possible outcomes must equal exactly 1, i.e., \(\sum P(x_i) = 1\).
We will examine each of the four provided tables based on these two criteria.
- Evaluate each table against the conditions
Let's evaluate each table:
- Table 1: Contains negative probabilities (\(-1\)). This violates the non-negativity condition.
- Table 2: All probabilities (\(0.11\), \(0.05\), \(0.49\), \(0.35\)) are non-negative. We calculate their sum:
\(0.11 + 0.05 + 0.49 + 0.35 = 1.0\). This satisfies both conditions.
- Table 3: Contains a negative probability (\(-0.49\)). This violates the non-negativity condition.
- Table 4: Contains a negative probability (\(-0.1\)). This violates the non-negativity condition. Furthermore, its sum is \(0.5 + 0.4 + 0.2 - 0.1 = 1.0\), but since it violates the first condition, it cannot be a valid probability distribution.
- Conclusion
Based on our evaluation, Table 2 is the only one that satisfies both the non-negativity condition (all \(P \geq 0\)) and the normalization condition (the sum is exactly \(1\)). Therefore, Table 2 is the correct probability distribution table.
Examples
Probability distributions are fundamental in statistics. For example, if you are designing a survey and assigning probabilities to different potential survey responses (like "Strongly Agree," "Agree," "Disagree," "Strongly Disagree"), you must ensure your assigned probabilities sum to 1 and are all non-negative to make valid statistical predictions. Applying these rules helps ensure that your model of reality is logically consistent.