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In probability, the "expected value" refers to the average amount one could expect to win or lose in a game of chance over numerous trials. It's a key concept that helps determine the fairness or profitability of games based on probabilistic outcomes.
When calculating the expected value, you multiply each possible outcome by its probability and then sum these values. In the context of our dice game, we are interested in winning when we roll doubles. The calculation becomes:

• Probability of rolling doubles: \( \frac{1}{6} \)
• Prize for rolling doubles: \( 5 \)
• Expected value = Probability \( \times \) Prize = \( \frac{1}{6} \times 5 = \frac{5}{6} \)

Thus, you can expect to "win" \( \frac{5}{6} \) dollars each time you play the game. This isn't what you receive in total but the average over many plays. A higher expected value suggests a more favorable game from a player's perspective.

A "fair game" in the context of probability means that the game's expected value of the outcomes is equal to the cost required to play. It ensures that neither the player nor the host has a mathematical advantage in the long run. This is crucial for games based on chance as it addresses the concept of fairness and incentives in gaming.
In our specific dice game scenario, to determine the fair cost to play:

• We calculate the expected value of winning, which is \( \frac{5}{6} \) dollars.
• The cost to play should also be \( \frac{5}{6} \) dollars.

When these values are equal, the game does not favor the player or the house over time. Every dollar paid to play should be equivalent to the average payout received. If the cost were higher, the game would be unfair against the player; if lower, unfair against the house.

Dice probability refers to the likelihood of various outcomes from rolling dice, a common way to introduce basic probability concepts. In our context, we are interested in the probability of rolling doubles with a pair of six-sided dice.
This calculation involves the following:

• Each die has 6 faces, hence 36 possible outcomes (6 faces on the first die \( \times \) 6 faces on the second die).
• Rolling doubles means both dice show the same number. Possible doubles are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6).
• The probability of rolling doubles is thus \( \frac{6}{36} = \frac{1}{6} \).

Understanding this probability helps assess how likely you are to win a game based on rolling doubles, guiding you in managing expectations and strategic decisions in gambling contexts. It highlights that while luck plays a role, the structure of probabilities determines the fairness and potential success of a gaming strategy.