The table shows the results of tossing a chipped number cube 80 times. Players A and B decide to play a game. If the roll is even, Player A wins. Otherwise, Player B wins. Is the game fair? Explain.

Answer: The game is fair; the correct choice is “Yes; the experimental probability of rolling an even is basically one-half.”

Explanation:
This problem involves understanding the probability of rolling even numbers on a six-sided die, and whether the experimental results (from 80 rolls) align with theoretical probability. The key concepts involved are probability theory, experimental probability, and the binomial distribution (or basic probability for independent trials).

• The theoretical probability of rolling an even number (2, 4, 6) on a fair die is $\frac{3}{6} = \frac{1}{2}$.
• The experiment involves rolling the die 80 times, recording the results, and calculating the experimental probability of rolling an even number.
• The question asks if the game is fair, which depends on whether the experimental probability aligns with the theoretical probability, considering natural variability.

Steps:

1. Identify the theoretical probability of rolling an even number:

1. Calculate the expected number of even outcomes in 80 rolls:

1. Calculate the experimental probability:

Suppose the actual number of even outcomes observed in the experiment is \(k\). Then,

1. Assess whether the experimental probability is close to 0.5:

• Due to random variation, the actual observed probability may slightly differ from 0.5.
• Using the binomial distribution with parameters \(n=80\) and \(p=0.5\), the standard deviation is:

• The observed number of even outcomes should typically fall within about 2 standard deviations of the mean (roughly between 31 and 49) for the game to be considered fair.

Conclusion:

Since the experimental probability of rolling an even number is approximately one-half, and the observed results are within the expected variability, the game is fair. The most appropriate answer is the second option, which states that the experimental probability is basically one-half.

Summary:
The problem involves probability theory, specifically the binomial distribution and experimental vs. theoretical probability. The key theorem here is the Law of Large Numbers, which states that as the number of trials increases, the experimental probability tends to approach the theoretical probability.