Question

in a game, the players spin a spinner with 6 equal sections numbered 1 through 6. the table shows three versions of the game with different sets of rules to decide who earns points on a spin. move the options to the spaces to show whether or not each version of the game is mathematically fair.

game	player 1 wins	player 2 wins	fair or not fair?
2	if the spinner shows 1 or 2.	if the spinner shows 3 through 6.
3	if the spinner shows 1 through 4.	if the spinner shows a 5 or 6.

options: fair, not fair (below the table)

Explanation:

To determine if each game is fair, we check if the probability of Player 1 winning equals the probability of Player 2 winning. The spinner has 6 equal sections (1–6).

Game 1

• Player 1 Wins: Odd numbers (1, 3, 5) → 3 outcomes.

Probability: $\frac{3}{6} = \frac{1}{2}$.

• Player 2 Wins: Even numbers (2, 4, 6) → 3 outcomes.

Probability: $\frac{3}{6} = \frac{1}{2}$.

• Since $\frac{1}{2} = \frac{1}{2}$, Game 1 is Fair.

Game 2

• Player 1 Wins: 1 or 2 → 2 outcomes.

Probability: $\frac{2}{6} = \frac{1}{3}$.

• Player 2 Wins: 3–6 → 4 outcomes (3, 4, 5, 6).

Probability: $\frac{4}{6} = \frac{2}{3}$.

• Since $\frac{1}{3}

eq \frac{2}{3}$, Game 2 is Not Fair.

Game 3

• Player 1 Wins: 1–4 → 4 outcomes.

Probability: $\frac{4}{6} = \frac{2}{3}$.

• Player 2 Wins: 5 or 6 → 2 outcomes.

Probability: $\frac{2}{6} = \frac{1}{3}$.

• Since $\frac{2}{3}

eq \frac{1}{3}$, Game 3 is Not Fair.

Final Table

Game	Player 1 Wins	Player 2 Wins	Fair or Not Fair?
2	1 or 2	3–6	Not Fair
3	1–4	5 or 6	Not Fair

(To complete the task, move "Fair" to Game 1’s "Fair or Not Fair?" column, and "Not Fair" to Games 2 and 3’s columns.)

Answer:

To determine if each game is fair, we check if the probability of Player 1 winning equals the probability of Player 2 winning. The spinner has 6 equal sections (1–6).

Game 1

• Player 1 Wins: Odd numbers (1, 3, 5) → 3 outcomes.

Probability: $\frac{3}{6} = \frac{1}{2}$.

• Player 2 Wins: Even numbers (2, 4, 6) → 3 outcomes.

Probability: $\frac{3}{6} = \frac{1}{2}$.

• Since $\frac{1}{2} = \frac{1}{2}$, Game 1 is Fair.

Game 2

• Player 1 Wins: 1 or 2 → 2 outcomes.

Probability: $\frac{2}{6} = \frac{1}{3}$.

• Player 2 Wins: 3–6 → 4 outcomes (3, 4, 5, 6).

Probability: $\frac{4}{6} = \frac{2}{3}$.

• Since $\frac{1}{3}

eq \frac{2}{3}$, Game 2 is Not Fair.

Game 3

• Player 1 Wins: 1–4 → 4 outcomes.

Probability: $\frac{4}{6} = \frac{2}{3}$.

• Player 2 Wins: 5 or 6 → 2 outcomes.

Probability: $\frac{2}{6} = \frac{1}{3}$.

• Since $\frac{2}{3}

eq \frac{1}{3}$, Game 3 is Not Fair.

Final Table

Game	Player 1 Wins	Player 2 Wins	Fair or Not Fair?
2	1 or 2	3–6	Not Fair
3	1–4	5 or 6	Not Fair

(To complete the task, move "Fair" to Game 1’s "Fair or Not Fair?" column, and "Not Fair" to Games 2 and 3’s columns.)