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Probability is a way of quantifying the likelihood of an event occurring. In the context of a dice game, understanding probability is crucial. Each die has 6 faces, numbered from 1 to 6. This means that each number has an equal chance of landing face up.

To find the probability of a specific face, such as a "1", you calculate it as the ratio of favorable outcomes (rolling a 1) to the total possible outcomes (all possible rolls). Thus, the probability of rolling a 1 is \(\frac{1}{6}\).

Similarly, the likelihood of not rolling a 1, termed "not 1", is the complement of rolling a 1. You compute this as \(1 - \frac{1}{6} = \frac{5}{6}\). These probabilities lay the foundation for determining how likely different results are in a game scenario.

In a game, the payoff is the reward you receive for a specific outcome. Calculating the payoff is essential for determining the player's financial results after several rounds. Each outcome's payoff is the result of the game's rules minus any costs associated with playing.

For instance, in this dice game, a player wins \(5 when rolling a 1, but they pay \)1 to play the game. The net payoff, in this case, is \(\\(5 - \\)1 = \\(4\).

Conversely, rolling any other number results in a net payoff of \(\\)0 - \\(1 = -\\)1\), as the player does not win any money but still incurs the playing cost. These calculated payoffs then feed into finding the game's expected value, giving insight into potential winnings or losses.

Understanding the likelihood of various outcomes helps players make informed decisions. In this instance, there are two possible outcomes: rolling a 1 or not rolling a 1. The probability of the first outcome—rolling a 1—is \(\frac{1}{6}\), while not rolling a 1 has a probability of \(\frac{5}{6}\).

By assigning probabilities to each outcome, players can anticipate the frequency of winning or losing. This is crucial in evaluating whether a game is worth playing repeatedly from a statistical perspective. Knowing these probabilities allows players to calculate the expected value, providing a numerical summary of potential financial outcomes over many games.

Game theory involves applying mathematical models to understand decision-making in competitive scenarios, like games. In this dice game, game theory principles can help determine whether a player should participate.

By leveraging calculations of probability and payoff, and summing them to find expected values, players assess the long-term benefits of playing.

The expected value formula, \(\text{Expected Value} = \text{Probability of Winning} \times \text{Payoff} + \text{Probability of Losing} \times \text{Loss}\), gives a blueprint of how much a player can expect to gain or lose per game on average. Negative or positive outcomes from this formula suggest whether the game is economically favorable or not.

Thus, game theory aids players in making strategically sound decisions in games of chance by weighing risks and rewards.