Compare the rules of two games. Game A: Two six-sided dice are thrown. The numbers on the dice determine which player gets a point. Player 1 gets a point if the product of the numbers on the dice is even. Player 2 gets a point if the product is odd. Game B: Two six-sided dice are thrown. The numbers on the dice determine which player gets a point. Player 1 gets a point if the sum of the numbers on the dice is even. Player 2 gets a point if the sum is odd. Which statement accurately describes whether these games are fair?
The Correct Answer and Explanation is:
Correct Answer:
Game A is not fair; Game B is fair.
Explanation (300 words):
To determine whether the games are fair, we must examine the probability of each player winning in both games. A game is fair if all players have an equal probability of winning, i.e., each has a 50% chance.
Game A: Product-Based Rules
- Player 1 scores if the product of the two dice is even.
- Player 2 scores if the product is odd.
Recall:
- An odd product occurs only if both dice roll odd numbers.
- The odd numbers on a die are 1, 3, and 5 — so each die has a 3/6 = 1/2 chance of being odd.
- The probability that both dice show odd numbers = (1/2) × (1/2) = 1/4.
Thus:
- Probability product is odd (Player 2 scores) = 1/4.
- Probability product is even (Player 1 scores) = 1 – 1/4 = 3/4.
➡️ Game A is not fair, because Player 1 has a 75% chance of scoring, while Player 2 only has 25%.
Game B: Sum-Based Rules
- Player 1 scores if the sum is even.
- Player 2 scores if the sum is odd.
Let’s examine the probabilities:
- A sum is even if both dice are even or both are odd.
- A sum is odd if one die is even and the other is odd.
Each die has 3 odd and 3 even numbers.
So:
- P(even, even) = (1/2) × (1/2) = 1/4
- P(odd, odd) = (1/2) × (1/2) = 1/4
- P(one even, one odd) = 1 – (1/4 + 1/4) = 1/2
Thus:
- Probability sum is even = 1/4 + 1/4 = 1/2
- Probability sum is odd = 1/2
➡️ Game B is fair, as both players have a 50% chance of winning.
Conclusion:
Game A is biased in favor of Player 1, while Game B gives both players an equal chance — it is fair.
