91Ó°ÊÓ

In probability theory, the expected value is a fundamental concept that gives a measure of the "center" of a probability distribution. It's essentially the average outcome you'd expect from a random experiment if you could repeat it infinitely many times. To calculate the expected value for our die-rolling game, we multiply each possible payout by its probability and sum these products. This results in:\[ \E = \frac{1}{6}(1) + \frac{1}{6}(3) + \frac{1}{6}(5) - \frac{1}{6}(2) - \frac{1}{6}(4) - \frac{1}{6}(6) \]This equation simplifies to:- for odd outcomes (1, 3, and 5): - \( \frac{1}{6}(1) = \frac{1}{6} \) - \( \frac{1}{6}(3) = \frac{1}{2} \) - \( \frac{1}{6}(5) = \frac{5}{6} \)- for even losses (2, 4, and 6): - \( \frac{1}{6}(2) = \frac{1}{3} \) - \( \frac{1}{6}(4) = \frac{2}{3} \) - \( \frac{1}{6}(6) = 1 \)After simplifying, each component contributes to an expected value of 0, indicating that over time, you'd neither win nor lose money on average.

A game is considered "fair" if the expected value of the payouts from that game is zero. In a fair game, players do not have an advantage or disadvantage. Over time, the average net gain for a participant is zero. In our exercise, we determine fairness by calculating the expected value of the game, which we found to be zero. This implies:

• There is no net gain or loss on average over many plays.
• The probabilities and payouts are balanced perfectly, so neither the player nor the house stands to systematically win.

Thus, given this evaluation, we conclude that the described die-rolling game qualifies as a fair game.

Understanding probability distribution is key to analyzing games of chance. A probability distribution lists all possible outcomes of an experiment and assigns a probability to each one.In our dice game, there are six possible outcomes when rolling a die:

• 1, where you win \\(1
• 2, where you lose \\)2
• 3, where you win \\(3
• 4, where you lose \\)4
• 5, where you win \\(5
• 6, where you lose \\)6

Each outcome has a probability of \( \frac{1}{6} \).By examining this probability distribution, compounded by the odds, you can effectively calculate the expected value and determine the fairness of the game. This systematic approach helps to decide if engaging in such a game is strategically advantageous or neutral in the long run.