A fair three-sided die (or equivalent mechanism like a spin wheel with three equal sectors) implies that the three outcomes $R$, $S$, and $P$ are equally likely. This ensures that the probabilities are evenly distributed.

Calculation:

Since the die is fair, the total probability is $1$, and each of the three outcomes has an equal chance:

$\text{Probability of each outcome} = \frac{1}{\text{Number of outcomes}} = \frac{1}{3}.$

Probability Distribution:

a) $Pr(R) = \frac{1}{3}$

b) $Pr(S) = \frac{1}{3}$

c) $Pr(P) = \frac{1}{3}$

Verification:

The sum of probabilities should equal $1$: $Pr(R) + Pr(S) + Pr(P) = \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1.$

Thus, the distribution is valid.

Would you like more details or clarifications? Here are some related questions to consider:

1. What is the probability of rolling either $R$ or $S$?
2. How would the probabilities change if the die were biased?
3. Can you generalize this method to find the probabilities for a $n$-sided die?
4. How would you simulate rolling a fair three-sided die using a standard six-sided die?
5. What are some real-life applications of probability distributions like this?

Tip: When working with probabilities, always check that the total probability adds up to $1$.