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In probability theory, understanding the range of valid values is essential. Probabilities indicate the likelihood of an event occurring and must always fall within a specific range. The valid range for probability values is from 0 to 1, inclusive. A probability of 0 means the event will never happen, while a probability of 1 indicates the event will definitely occur.
To illustrate, if an event has a probability of 0.5, it means the event has a 50% chance of occurring. Simply put, any number outside this range, like -0.5 or 1.5, cannot represent a probability.
Understanding this range helps in evaluating and interpreting probabilities accurately.

When evaluating if a given number could be a valid probability, it's crucial to check if it falls within the range of 0 to 1. Let's go through the numbers provided:
• **1.5**: This number is greater than 1 and thus cannot be a probability.
• **\( \frac{1}{2} \)**: Also known as 0.5, this value lies within the range, making it a valid probability.
• **\( \frac{3}{4} \)**: Equivalent to 0.75, which is within our acceptable range.
• **\( \frac{2}{3} \)**: Approximately 0.6667, this value is valid as it falls between 0 and 1.
• **0**: At the very boundary of our range, it's still an acceptable probability.
• **-\( \frac{1}{4} \)**: Negative values are not valid as probabilities and thus this number is invalid.
Clearly, the only valid probabilities in our list are \( \frac{1}{2} \), \( \frac{3}{4} \), \( \frac{2}{3} \), and 0.

Breaking down the problem into steps ensures we grasp the concept fully.
**Step 1 - Understand the Range**: As covered earlier, probabilities must be between 0 and 1, inclusive.
**Step 2 - Evaluate Each Number**: Examine the given numbers and which of them fall in the range. Here are the numbers: 1.5, \( \frac{1}{2} \), \( \frac{3}{4} \), \( \frac{2}{3} \), 0, and -\( \frac{1}{4} \).
**Step 3 - Check 1.5**: Since 1.5 is greater than 1, it can't be a probability.
**Step 4 - Check \( \frac{1}{2} \)**: 0.5 is a valid probability because it falls within 0 to 1.
**Step 5 - Check \( \frac{3}{4} \)**: With a value of 0.75, it falls within the range.
**Step 6 - Check \( \frac{2}{3} \)**: Roughly 0.6667, another valid probability.
**Step 7 - Check 0**: As 0 is the lower limit, it’s valid.
**Step 8 - Check -\( \frac{1}{4} \)**: Being negative, it's not valid.
By following these steps, you can determine the validity of probability values effectively.