Final answer:
To determine if a game is fair, you calculate the expected value. For the scenario of tossing a biased coin where P(heads)=3/4 and P(tails)=1/4, the expected value is negative, suggesting you will lose money over time. Thus, this game is not fair to the player. EV = (\frac{3}{4} \times -$6) + (\frac{1}{4} \times $10) = -$4.50 + $2.50 = -$2.00
Step-by-step explanation:
To determine whether the game at the fair is fair, we would need to know the probability of landing on a shaded face of the dice. Since that information is not provided, we'll consider a different scenario with dice and coins for comparison.
When assessing the fairness of a game, we calculate the expected value, which is the average amount of money one can expect to win or lose per play if the game is played many times. Fairness in a game means that the expected value is zero, which indicates that, in the long run, you neither gain nor lose money. If the expected value is positive, you will come out ahead; if negative, you will lose money over many plays.
Let's examine the game of tossing a biased coin with probabilities P(heads) = \frac{3}{4} and P(tails) = \frac{1}{4}. The expected value (EV) for this game can be calculated using the formula:
EV = (P(win) \times Amount won per win) - (P(lose) \times Amount lost per loss)
For the biased coin, this would be:
EV = (\frac{3}{4} \times -$6) + (\frac{1}{4} \times $10) = -$4.50 + $2.50 = -$2.00
Since the expected value is negative, you would expect to lose money over time, and thus the game is not fair to the player.