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When we talk about probability in games of chance, we're exploring how likely certain outcomes are. Take the roll of a die, for instance. A standard die has six equally likely outcomes, ranging from 1 to 6. Since all faces have an equal chance of landing face up, the probability of any single outcome is one out of six, or \( \frac{1}{6} \).

In the context of our game, the chances of rolling a 1, 2, or 3 are the same as rolling a 4, 5, or 6. However, the impact of these rolls is not the same due to the rules of the game. Understanding the probability of each outcome is the foundation upon which we assess a game's fairness or the expected results over time. It's worth noting that probability does not predict individual events but gives a measure of the likelihood of different results over many repetitions.

A random variable is a numerical description of the outcome of a statistical experiment, such as rolling a die in our game. Each potential outcome of the experiment corresponds to a different value of the random variable. For instance, rolling a 1, 2, or 3 on the die can be seen as random variables that result in winning \( \$1 \), \( \$2 \), and \( \$3 \), respectively.

In the given game, we have a random variable that represents the net gain or loss of the player for each roll. Since the outcomes are directly related to the numbers on the die, this random variable converts the non-numeric outcome of a die roll into a monetary gain or loss. It's an essential concept for translating the abstract notions of chance into concrete numerical values that can be analyzed and predicted.

Mathematical expectation, or expected value, is a critical concept when assessing games of chance. It represents the average amount one can expect to win or lose per bet if the bet is repeated many times. The expected value is calculated by multiplying each possible outcome by its associated probability and then summing all these values.

In this game, the expected value reflects the average monetary return for each game played, taking into account the cost to play and the possible winnings. It's determined by considering the amount won or lost for each roll of the die and then accounting for the probability of each of those rolls occurring. The results tell us whether the player is likely to gain money over time or lose it. In our example, with the expected value being less than zero, it suggests that the game, on average, is a losing proposition for the player when played multiple times.