Question

Assessing the fairness of a board game with the given probability distribution:

A. It's not a fair game because the weighted average is negative.
B. It's a fair game because you could win points.
C. It's a fair game because you gain more points if you roll a 5 than if you roll any of the negative outcomes.
D. It's not a fair game because the weighted average is positive.

Answer 1

To assess the fairness of a board game, the expected value must be calculated, which should be zero for a fair game. The fairness cannot be assessed by the potential to win points or larger rewards for specific outcomes without considering the overall weighted average of all possible outcomes.

Step-by-step explanation:

When assessing the fairness of a board game, one must consider the expected value, which is the long-term average result of the game as the number of repetitions becomes large. To determine if a game is fair, calculations must be performed to find the weighted average, or expected value, of all possible outcomes. In a fair game, this value should be zero, indicating that over time, players will neither gain nor lose money. If the weighted average is negative, the player is expected to lose money over time, making it an unfair game. Conversely, if the weighted average is positive, the player would typically expect to gain money over time, which might also reflect an unfair game if the context implies that the game should have a neutral expected value.

For the given board game, saying that it's a fair game because you could win points (option B) or because you gain more points for a specific outcome (option C) is not sufficient to establish fairness. The most accurate assessment would depend on calculating the expected value. If the expected value is negative (option A), the game is not fair because players will lose money on average. If the expected value is positive (option D), the game may be considered unfair if the expectation is that neither the player nor the house should have the advantage. Therefore, without numerical values to calculate an expected value, it is impossible to definitively state whether the game is fair or not, based solely on the information given in the options.

Answer 2

The fairness of a board game can be assessed by calculating the expected value, which if zero, denotes a fair game. When given probabilities and outcomes, like the biased coin or card games described, one calculates expected value to determine potential long-term average earnings, which in turn indicates if the game is fair.

Step-by-step explanation:

The fairness of a board game or any game involving probabilistic outcomes can often be assessed using the concept of expected value, which is the weighted average of all possible outcomes. To determine if the game is fair, we calculate the expected value by multiplying each outcome by its probability and then summing these products. If the expected value is zero, the game is considered fair because the average winnings over time would neither gain nor lose money. If the expected value is negative, you would expect to lose money over time, and if it is positive, you would expect to gain money over time.

In the case of the coin with a 56 percent chance of landing on heads and a 44 percent chance of landing on tails, if the payouts were equal for both sides, the expected value would not be zero. More trials would help determine the fairness with greater certainty. For the die-rolling game, one needs to account for the probability of rolling at least a five, which is 2 out of 6, or 1 out of 3.

For the card and coin game, the expected value would require calculating the probabilities of drawing a face card (12 out of 52) and the results of the coin toss (50 percent for both heads and tails), along with the corresponding wins or losses. Similarly, the game with the biased coin needs the probabilities and payouts accounted for to determine if one would come out ahead over many plays.

Ultimately, knowing the expected value helps in making decisions to play or not based on whether the long-term average expected earnings are positive or negative.