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A Random Variable is a fundamental concept in probability and statistics. It essentially represents a variable whose possible values are numerical outcomes of a random phenomenon. In the context of our exercise, the random variable is denoted as \(X\), and it represents the least (or smallest) number on the card obtained when \(m\) cards are drawn from \(n\) cards without replacement.

Random variables can be discrete or continuous. Discrete random variables take on a fixed number of distinct values, while continuous variables can take on an infinite number of values within a range. Here, \(X\) is a discrete random variable because the least number on drawn cards takes on integer values between 1 and \(n\).

• Outcome of Random Variable: Each possible outcome of \(X\) is the smallest number in the drawn set of \(m\) cards.
• Example: If you have cards numbered 1 to 5, and you draw 3 cards such as {2, 3, 5}, the random variable \(X\) would be 2.

The Cumulative Distribution Function (CDF) of a random variable is a critical tool that helps in understanding the distribution of the random variable by providing the probabilities that \(X\) is less than or equal to a certain value. This gives us a complete picture of the distribution of probabilities.

In our exercise, the CDF is noted as \( F(k) = P(X \leq k) \). It is calculated by summing up the probabilities for all outcomes less than or equal to \(k\). This gives us the combined probability of \(X\) reaching any of those lower or equal values.

• Usage: CDF is especially useful because it can tell you the likelihood of the random variable being less or equal to any given threshold.
• Easy Interpretation: By looking at \( F(k) \), you can easily assess how often values equal to or less than \(k\) appear in multiple draws.
• Example: If \(X\) can take the values 1, 2, and 3, and each has probabilities 0.2, 0.3, and 0.5 respectively, \( F(2) \) would be 0.2 + 0.3 = 0.5.

Combinatorics is often used in probability calculation. It's a branch of mathematics focusing on the counting, arrangement, and combination of objects. In this problem, it plays a key role in determining the number of ways we can draw cards, which is essential for calculating probabilities.

To find the probability distribution of our random variable \(X\), combinatorics are used to calculate the different possible combinations of drawing cards. For example, when calculating \( P(X = k) \), we use combinatorics:

• Numerator: \(\binom{n-k}{m-1}\) - This represents how many ways there are to draw the remaining \(m-1\) cards from the list of \(n-k\) cards not including the smallest card.
• Denominator: \(\binom{n}{m}\) - Total ways to draw \(m\) cards from \(n\) cards.

These combinations help us determine precise probabilities by comparing how many ways an event can happen versus the total possibilities.