Question

In a board game, you roll a die to win or lose points, depending on the outcome. the outcomes follow this probability distribution. number 1 5 2,3,4,6 probability 1/6 1/6 4/6 points 4 6 -5 if a player can only win by accumulating 20 points, which of the following best describes the fairness of the game?

Options:
a. It's not a fair game because the weighted average is negative.
b. It's not a fair game because the weighted average is positive.
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 a fair game because you could win points.

Answer 1

The fairness of the game is evaluated through the expected value, which is negative in this case indicating that a player tends to lose points over time. Therefore, it is not a fair game.

Step-by-step explanation:

To determine the fairness of the game described, we need to calculate the expected value. This will help us understand whether a player can expect to gain or lose points, on average, over a large number of games.

According to the probability distribution given:

• Rolling a 1 gives you 4 points with a probability of 1/6.
• Rolling a 5 gives you 6 points with a probability of 1/6.
• Rolling a 2, 3, 4, or 6 loses you 5 points each with a combined probability of 4/6.

The expected value (EV) is calculated as follows:

EV = (1/6 × 4) + (1/6 × 6) + (4/6 × -5) = 2/3 + 1 - 10/3 = -1.67 (approximately)

Since the expected value is negative, it indicates that over time, a player is more likely to lose points than gain. Therefore, the correct answer to the original question is:

a. It's not a fair game because the weighted average is negative.