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A random variable is a function that assigns numerical values to the outcomes of a random experiment. In the context of our coin tossing experiment, we define a discrete random variable, denoted as \(X\), which counts the number of heads resulting from two tosses of a coin.

A discrete random variable, like \(X\), takes on a finite or countably infinite set of values. It differs from a continuous random variable, which can take any value within a given range.

For our experiment with two coin tosses, the possible values for \(X\) are 0, 1, and 2 heads. These values are countable, making \(X\) a discrete random variable. Discrete random variables often arise in scenarios involving counting, such as tallying specific outcomes.

Understanding whether a random variable is discrete or continuous helps us choose the right methods to analyze and interpret data.

In probability and statistics, experimental outcomes are the possible results of an experiment. For our coin-tossing experiment, each set of outcomes is an arranged sequence of two coin tosses.

The coin can land in two orientations for each toss: heads (H) or tails (T). Therefore, tossing a coin twice results in four possible experimental outcomes:

• Heads-Heads (HH)
• Heads-Tails (HT)
• Tails-Heads (TH)
• Tails-Tails (TT)

Cataloging these outcomes is crucial because it serves as the foundation for defining random variables and calculating probabilities.

Each outcome aligns with specific numerical values of our random variable \(X\), based on the number of heads observed.

Probability is the measure of the likelihood that an event will occur as a result of a chance process. In the coin-tossing experiment, each experimental outcome has an equal likelihood since the coin is fair.

The probability of each outcome for two tosses is calculated as \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\), since each toss is independent. Thus, each of the outcomes HH, HT, TH, and TT has a probability of \(\frac{1}{4}\).

Understanding probabilities helps us predict the outcomes of experiments over time. It allows us to analyze the behavior of our discrete random variable \(X\) by assigning probabilities to its potential values. For example, the probability that \(X = 0\) (no heads) is the probability of the TT outcome, which is \(\frac{1}{4}\). Similarly, the probability that \(X = 1\) (one head) combines the probabilities of HT and TH, which is \(\frac{1}{4} + \frac{1}{4} = \frac{1}{2}\).