The question asks which statement must be TRUE for a fair random number generator that generates numbers on the interval $[0, 1]$. Let's go through each option:

A. Each number between 0 and 1 is equally likely to occur.

• This describes a uniform distribution, which is typical for a fair random number generator over a continuous interval. In this case, every number between 0 and 1 should have an equal probability of being generated. This is a true statement for a fair generator.

B. The numbers 0 and 1 will not occur.

• For a random number generator on the interval $[0, 1]$, whether 0 and 1 are included depends on the specific design. In practice, generators usually include 0 and exclude 1 (i.e., $[0, 1)$), but they could technically generate both. This is not necessarily true.

C. There should be more numbers generated around 0.5 than around 0 or 1, because 0.5 is the median.

• In a uniform distribution, all numbers are equally likely, so there wouldn't be more numbers around 0.5. This statement would be true if the distribution were normal or bell-shaped, but for a uniform distribution, it's false.

D. Each number between 0 and 1 must have a different probability of occurring.

• This contradicts the idea of a uniform distribution, where every number in the interval has the same probability. So, this statement is false.

Conclusion:

The statement that must be true for a fair random number generator is A: "Each number between 0 and 1 is equally likely to occur."

Would you like further details or explanations? Here are five related questions you might want to consider:

1. What is the difference between uniform and normal distributions?
2. How do random number generators work computationally?
3. Why might some random number generators include 0 but not 1?
4. What are the properties of a fair random number generator?
5. How can the fairness of a random number generator be tested?

Tip: When dealing with continuous random variables, remember that in a uniform distribution, the probability density function (PDF) is constant across the interval.