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In probability theory, when dealing with joint distributions, marginal distribution is an efficient way to simplify the model by focusing on one particular variable or component and assessing its behaviour independently. Essentially, the marginal distribution of a single random variable can be found by summing (or integrating, for continuous cases) the joint probabilities over all other variables. This helps us understand the distribution of one variable without considering the influence of others in a complex dataset.

To illustrate this, let's consider the marginal distribution for our committee selection problem. The marginal distribution of Republicans, represented by \(Y_1\), is found by summing all joint probabilities across various values of Democrats \(Y_2\). Using the step-by-step approach from the exercise, we process:

• For \(P(Y_1 = 0)\), sum probabilities for all \(Y_2 = 0, 1, 2\).
• Similarly, compute \(P(Y_2 = j)\) for each differing scenario \(Y_1 = 0, 1, 2, 3\).

Thus, this method gives a clear distribution of one type of political affiliation in the committee independent of the presence of other affiliations.

Conditional probability allows us to determine the likelihood of an event (say, \(A\)) given that another event (\(B\)) is already known to have occurred. This is pivotal when we want to make informed predictions based on existing information. It's defined by the formula: \(P(A | B) = \frac{P(A \cap B)}{P(B)}\).

In the context of the exercise, consider using conditional probability to find \(P(Y_1 = 1 | Y_2 \geq 1)\), i.e., the probability that the committee includes exactly one Republican given that there is at least one Democrat. This requires:

• Calculating \(P(Y_1 = 1 \cap Y_2 \geq 1)\), where committees have 1 Republican and at least 1 Democrat (either 1 or 2 Democrats).
• Calculating \(P(Y_2 \geq 1)\), the probability that there's at least 1 Democrat.

Once these probabilities are evaluated, use the formula to obtain the conditional probability.

In probability and statistics, combinations are critical for determining how many ways a subset of items can be selected from a larger set, without regard for the order of selection. The formula to calculate combinations, or "n choose k," is: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

The formula allows us to calculate the number of ways \(k\) items can be selected from \(n\) items and plays a central role when determining probability distributions in scenarios like committee selections. In the given exercise, applying the combination formula helps to calculate the different ways to form a committee:

• First, compute the total possible committees, \(\binom{9}{3}\), from nine people.
• Then, each specific scenario can be mapped, like finding \(\binom{4}{1}\) for one Republican, \(\binom{3}{1}\) for one Democrat, and subsequently find combinations for other scenarios.

Utilizing this formula allows us to systematically account for all possible committee configurations and derive the joint probabilities required to solve the problem.