Final answer:
The probabilities for states of nature must be between 0 and 1 and must sum to 1 (option a), following the basic principles of probability where the sum of the probabilities of a success (p) and a failure (q) equals 1, which is true for repeated independent trials.
Step-by-step explanation:
Understanding the Probabilities for States of Nature
The probabilities for the states of nature must be between 0 and 1, and their sum must equal 1. This is the fundamental principle of probability, which states that the probability (p) of a success and the probability (q) of a failure in any experiment are complementary, meaning that p + q = 1 and q = 1 - p.
For example, if we throw a fair die, the probability of rolling a three is \(rac{1}{6}\), and this probability remains constant no matter how many times we throw the die. If two events are independent and mutually exclusive, their joint probability is calculated by multiplying the individual probabilities, and their combined probability can be obtained by addition, following the basic rules of probability.
It's important to understand that the total probability of all possible outcomes in a sample space must add up to 1, as this reflects the certainty that one of the possible outcomes will occur. Furthermore, the probability of any single event cannot exceed 1 (or 100 percent), which signifies absolute certainty, nor can it be negative, as that would not have a meaningful interpretation in the context of probability theory.