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When dealing with the selection of items or individuals from a group, one often encounters the concept of combinations. A combination is a selection of items without considering the order. In simpler terms, when you want to choose a subset from a larger set, and the order does not matter, you use combinations.

To calculate the number of combinations possible, you can use a mathematical formula. For a set of size \( n \) and a choice of \( r \) items, the formula is given by:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

In this exercise, we have 4 applicants, and we need to choose 2. Therefore, the number of combinations is \( \binom{4}{2} = 6 \). This is why there are 6 different combinations or possible outcomes when choosing 2 applicants from 4.

Random selection is a fundamental concept in probability, which ensures each possible outcome has an equal chance of being chosen. This concept is pivotal in experiments where fairness is crucial, such as in the selection of applicants for a job or role.

By selecting applicants randomly, we ensure there's no bias or predisposition towards any candidate. This means each pair of applicants has the same likelihood of being chosen. In this exercise, each combination of applicants has a probability of \( \frac{1}{6} \) because there are 6 possible outcomes and each one should be equally likely.

It is important for students to understand that random selection forms the basis of fair decision-making processes, where each candidate or item has no more advantage over the other.

Selecting a member of a minority group in a random process highlights how probability can measure the chances of specific events happening. In this scenario, there is one minority member among four applicants. The concern is to calculate the probability of this particular person being selected.

The outcomes are examined to identify which ones include the minority group member, referred to as "D". These favorable outcomes are (A, D), (B, D), and (C, D). Each of these results in the minority member being chosen for one of the positions.

When you look at these 3 favorable results out of the total 6 outcomes, assigning an equal probability \( \frac{1}{6} \) to each means the total probability of selecting the minority group member becomes \( \frac{3}{6} \), simplifying to \( \frac{1}{2} \). This shows there's a 50% chance that the minority applicant will be hired when two candidates are chosen at random.

In probability, a sample space represents all the possible outcomes of a random experiment. This is the complete set of events that could possibly occur.

For this exercise, the sample space is composed of all the combinations of pairs of applicants that could be selected from the four available (A, B, C, D). Specifically, the sample space is:

• (A, B)
• (A, C)
• (A, D)
• (B, C)
• (B, D)
• (C, D)

Understanding the sample space is crucial because it lays the foundation for determining the probability of any particular event occurring. Events are simply subsets of the sample space. Therefore, being able to identify and enumerate all the possible outcomes is an essential step in solving any probability problem.