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In probability, a *simple event* is an outcome or result that cannot be broken down any further. Think of it as the building blocks in a probability scenario. In our cola experiment, each simple event is an arrangement of how the glasses of cola can be ranked in order of preference. Even if all the colas are identical, the person can still rank them in various sequences. For instance, consider the following permutations:

• (C, D, P)
• (C, P, D)
• (D, C, P)
• (D, P, C)
• (P, C, D)
• (P, D, C)

Each arrangement represents a simple event. It’s important to remember that a simple event pertains to the result of a single trial, meaning here it represents one specific order that the person could rank the glasses. Understanding simple events helps to break down complex problems into understandable parts and assists in calculating probabilities.

Permutations are essential when dealing with arrangements. They represent the possible ways to order a set of items. In our scenario, permutations help determine how the three glasses of cola might be ranked.When calculating permutations, the order of the items matters. For three items (our colas labeled C, D, and P), the number of permutations is calculated using the factorial method, denoted as \( n! \), where \( n \) is the total number of items. So for our exercise with three items: \[3! = 3 \times 2 \times 1 = 6\]This result aligns with the six different possible arrangements listed earlier. Recognizing the permutations of a set is crucial, as it lays the groundwork for determining the probabilities of different outcomes, especially when order matters as it does in ranking or listing preferences.

Assigning probabilities is all about understanding the chances of different outcomes occurring. In the cola ranking exercise, we start by assuming that each permutation, or order, is equally likely due to the identical nature of the colas.Since we identified 6 simple events (or permutations), and each one is equally probable, the probability for each permutation can be calculated as follows:\[\text{Probability of each permutation} = \frac{1}{6}\]This equal distribution assumes there is no bias or prior preference for any glass, due to them being identical. To handle specific questions like 'the probability of *C* being ranked first,' consider only those permutations where *C* holds the first position. From our list, these are:

• (C, D, P)
• (C, P, D)

The probability of *C* being first is:\[\frac{2}{6} = \frac{1}{3}\]For a more targeted question like the probability of *C* being first and *D* last, the permutation is uniquely defined as (C, P, D). So, the probability becomes:\[\frac{1}{6}\]Mastering how to assign probabilities ensures that we can answer any related probability questions with clarity and precision.