91Ó°ÊÓ

A random variable is a fundamental concept in probability and statistics, used to describe possible outcomes of a random process. It is essentially a variable whose possible values are numerical outcomes of a random phenomenon.
In this particular exercise, we define the random variable \( X \) as follows:

• \( X = 0 \) if the numbers rolled on both dice are the same.
• \( X = 1 \) if the numbers rolled on the two dice are different.

This transformation of actual outcomes into numerical values allows us to perform mathematical analysis and gain insights into the likelihood of various events occurring during the experiment.
By quantifying the results of the dice roll with a random variable, we can easily compute probabilities related to different scenarios, such as the chances of the dice showing the same or different numbers.

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes. It is a crucial tool for specifying the possible outcomes of a random variable and their corresponding probabilities.
In this exercise, we have a probability distribution established for the random variable \( X \), where:

• \( X = 0 \) with a probability of \( \frac{1}{6} \).
• \( X = 1 \) with a probability of \( \frac{5}{6} \).

This distribution is derived from the 36 possible outcomes when two dice are rolled. Calculating probabilities in this way helps us to understand the likelihood of each event in the context of our experiment.
By using this probability distribution, we can predict and infer the behavior over many trials, allowing us to make informed decisions or predictions about future experiments.

When dealing with dice, understanding the outcomes is important for calculating probabilities accurately. In this context, each die has six faces, numbered from 1 to 6. Hence, when two dice are rolled, there are \( 6 \times 6 = 36 \) possible combinations or outcomes.
The outcomes can be grouped into two scenarios depending on whether the dice show the same number or different numbers:

• Same Numbers: Pairs such as \((1, 1), (2, 2), \, \ldots \, , (6, 6)\) account for these outcomes. There are exactly 6 such pairs.
• Different Numbers: All remaining outcomes fall into this category and sum up to 30 different combinations.

The calculation of these outcomes forms the basis of determining probabilities for our random variable \( X \). Understanding dice outcomes is not only crucial in this exercise but also a cornerstone concept in many probability-related problems.