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In probability theory, the concept of a "sample space" serves as the foundational backbone for any probabilistic experiment. A sample space is essentially a set that encompasses all the possible outcomes that can occur in a specific experiment. For example, when considering the situation where three patients are choosing from three service stations at a hospital clinic, the sample space includes every conceivable combination of station selections these patients might make.
The notation typically employed represents these combinations in ordered sequences. Here, each patient's choice of station can be represented by the numbers 1, 2, and 3, corresponding to each station. This gives rise to possible sequences such as (1,1,1), (2,2,2), and so forth. In this exercise, since there are 3 patients and each can select from 3 stations, the total number of combinations possible is denoted by the formula \[ 3^3 = 27 \]This formula results from the choices available to each patient, multiplied across the three individuals.
Comprehending sample spaces is crucial because they lay the groundwork for defining events, calculating probabilities, and discerning the structure of complex experiments in probability theory.

Event probability, in the context of this exercise, is the measure of how likely a specific event is to occur within the sample space. When you have a clearly defined sample space composed of equal-probability outcomes, the probability of any event becomes the ratio of favorable outcomes to the total number of possible outcomes.
In our hospital clinic scenario, let event \(A\) be defined as each station receiving at least one patient. A key observation here is understanding which outcomes make event \(A\) possible. Suppose one outcome is (1,2,3), meaning each patient chooses a different station. Every arrangement of such a sequence constitutes a favorable outcome for \(A\). Since permutations like (1,2,3), (1,3,2), (2,1,3), (2,3,1), and so on, satisfy our condition, we enumerate six such permutations.
Assigning equal likelihood to each sequence means each sample point has a probability of:\[\frac{1}{27}\]Thus, the probability of event \(A\) happening can be found by summing the probabilities of all favorable outcomes:\[P(A) = 6 \times \frac{1}{27} = \frac{2}{9}\]Grasping event probability helps in predicting outcomes, modeling real-world uncertainties, and making informed decisions based on probable scenarios.

Permutations are concerned with the arrangement of a set of items where order matters. This concept is central to understanding various problems in probability and combinatorics, especially when dealing with distinct outcomes where sequence plays a role.
In our specific exercise, permutations help us determine the different ways patients can line up at the stations such that each station receives exactly one patient. For a given set like \(\{1,2,3\}\), a permutation examines every possible order in which the elements can appear. The formula for permutations of a set with \(n\) distinct elements is:\[ n! \]For the set \(\{1,2,3\}\), this results in:\[ 3! = 3 \times 2 \times 1 = 6 \]This explains why there are 6 different ways the patients can choose the stations non-repeatedly in this specific example.
Understanding permutations is invaluable when determining arrangements and allocations in probability, where the placement and order of selections significantly influence the outcome being analyzed.