91Ó°ÊÓ

Combinations are a way of selecting items from a larger pool where the order does not matter. In our exercise, we are concerned with choosing two students from a group of five. We need to select these students in pairs, and the order in which they are paired does not change the result. This is a classic example of combinations in mathematics. The formula to calculate combinations is represented as \( \binom{n}{r} \), where \( n \) is the total number of items to pick from, and \( r \) is the number of items to pick. For this problem, it's \( \binom{5}{2} \), which calculates to 10 possible pairs. Here are the potential pairs:

• (Abby, Deborah)
• (Abby, Mei-Ling)
• (Abby, Sam)
• (Abby, Roberto)
• (Deborah, Mei-Ling)
• (Deborah, Sam)
• (Deborah, Roberto)
• (Mei-Ling, Sam)
• (Mei-Ling, Roberto)
• (Sam, Roberto)

Understanding combinations helps us to solve the exercise efficiently and know all possible outcomes.

The sample space is a term used in probability to describe the set of all possible outcomes of an experiment or random process. In the context of this exercise, the sample space consists of all potential student pairs that could be drawn from the seminar course. Since there are five students, the number of combinations, as we calculated, is 10. Therefore, these 10 pairs represent our sample space. This is important because having a complete sample space allows us to see all potential outcomes and thus correctly determine probabilities. When looking at the full collection of possibilities, it ensures that nothing is missed which could skew our probability calculations later on.

A Simple Random Sample (SRS) is a method of selecting a subset of individuals from a larger population, such that each individual is chosen randomly and entirely by chance, giving each individual an equal probability of being selected at any stage during the selection process. In the context of our exercise, the two students are selected by drawing names from a hat, which perfectly exemplifies an SRS. This method ensures fairness because every pair of students has an equal chance of being selected, which is reflected in each having the same probability when calculated. The SRS is fundamental in statistics to ensure unbiased representation of a population, avoiding favoritism or sampling bias.

Probability calculation is about determining how likely an event is to occur. In this exercise, each outcome within the sample space is equally likely. Therefore, the probability of any particular pair being chosen is calculated by dividing 1 by the total number of possible outcomes: \( \frac{1}{10} \). For more specific probabilities, such as Mei-Ling being chosen, we count the number of favorable outcomes (those that include Mei-Ling) and divide by the total number of outcomes: \( \frac{4}{10} = \frac{2}{5} \). Similarly, for calculating the probability that both selected students liked the course, we count combinations where both students liked the course, giving us: \( \frac{3}{10} \). Understanding how to calculate these probabilities enables us to answer different parts of the problem accurately.