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When we pull items from a group and don't put them back, it's called sampling without replacement. This method is different from sampling with replacement, where each item could be selected more than once. Sampling without replacement makes the scenario more realistic for situations like drawing cards without returning them, or, as in our exercise, selecting firms from a group without the chance to pick the same firm twice.
In practice, sampling without replacement affects our probabilities. The number of choices changes as each item is selected and not returned. As a result, the probability distribution is not uniform, meaning each item isn't equally likely to be chosen with every selection. Instead, the probabilities adapt based on what's left in the pool.
In the exercise example, once a firm is selected, it cannot be chosen again, narrowing down the selection for subsequent firms. With six original firms and three being selected, the behavior of the probability changes with each step.

Combinatorial probability involves counting the number of successful outcomes over the total number of possible outcomes. It is a fundamental concept when working with hypergeometric distribution, which is used to determine probabilities in scenarios like our exercise, where items are drawn without replacement.
To calculate a combinatorial probability, we first need to identify the total number of combinations of drawing a set number of items, as shown in the problem using the combination formula \( \binom{n}{r} \). For example, choosing 3 firms out of 6 can be calculated by \( \binom{6}{3} = 20 \).
Next, we calculate the number of favorable outcomes for each scenario; e.g., drawing 1 nonlocal firm and 2 local ones requires calculating combinations for each step: \( \binom{2}{1} \) nonlocal times \( \binom{4}{2} \) local, which results in 12 specific combinations. The combinatorial probability is then the ratio of successful combinations to total combinations, such as \ \frac{12}{20} = 0.6 \ for having exactly one nonlocal firm.

Probability theory is the branch of mathematics dealing with the analysis of random events. The aim is to measure how likely an event will occur using different probability models. In this exercise, the hypergeometric distribution is used to calculate probabilities as we're dealing with sampling scenarios without replacement.
Within probability theory, there are important foundational concepts, including:

• Events and Outcomes: An event can be a single outcome or a group of outcomes, and probability helps in quantifying the likelihood of these occurring.
• Complementary Events: These are events that cover all possible outcomes when combined, such as having at least one nonlocal firm or having none in our exercise.
• Probability Complements: Probability of an event happening plus the probability of it not happening always equals 1; e.g., \(P(Y \geq 1) = 1 - P(Y = 0)\).

Probability theory helps us conceptualize and solve real-world problems by quantifying uncertainty and allowing us to make informed predictions or decisions.