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When dealing with probability, a fundamental concept that provides the basis for any analysis is the sample space. A sample space is the complete set of all possible outcomes of a probability experiment. In our exercise, the sample space involves three family members visiting a clinic with three stations available for each one.

Each family member, labeled as A, B, and C, can independently be assigned to one of the three stations numbered 1, 2, or 3. As there are 3 choices for A, 3 choices for B, and 3 choices for C, the total number of possible outcomes is a result of multiplying these choices together: \[3 \times 3 \times 3 = 27\] outcomes.

This forms the sample space, which includes outcomes such as:

• (1,1,1), which means A, B, and C all go to station 1
• (1,2,3), where A goes to station 1, B goes to station 2, and C goes to station 3
• (3,2,1), where A goes to station 3, B to station 2, and C to station 1

The full list of all 27 outcomes is systematically constructed by considering every possible combination, embracing the complete range of possibilities under the given conditions.

Permutations come into play when considering the different possible arrangements of a set of items. In our case involving the family, permutations are used when we want every member to go to a different station.

For three members going to three stations, each member must visit a unique station. This constitutes arranging three distinct items in three spaces. So, we calculate the number of permutations of three items, which happens to be \[3! = 3 \times 2 \times 1 = 6\] permutations.

These permutations are:

• (1,2,3): A assigned to 1, B to 2, C to 3
• (1,3,2): A to 1, B to 3, C to 2
• (2,1,3): A to 2, B to 1, C to 3
• (2,3,1): A to 2, B to 3, C to 1
• (3,1,2): A to 3, B to 1, C to 2
• (3,2,1): A to 3, B to 2, C to 1

The permutations give us all the possible ways to arrange A, B, and C across the stations, where no two family members will share the same station.

An event is a specific set of outcomes from the sample space that we are interested in examining. In context, an event might be something like all members visiting different stations, or none of them going to a specific station.

Let's focus on one interesting event: no member visiting station 2. For this to occur, each family member must be assigned to the remaining stations 1 or 3. This narrows down the potential outcomes as:

• (1,1,1): Everyone at station 1
• (1,1,3): A and B at station 1, C at station 3
• (1,3,1): A and C at station 1, B at station 3
• (1,3,3): A at station 1, B and C at station 3
• (3,1,1): B and C at station 1, A at station 3
• (3,1,3): B at station 1, A and C at station 3
• (3,3,1): C at station 1, A and B at station 3
• (3,3,3): Everyone at station 3

Each outcome meets the criteria of the event, displaying the flexibility and utility of understanding events within sample spaces. Events help in targeting specific conditions or probabilities we wish to calculate.