At a County fair there is a spinner game with 12 sectors: 2 red sectors, 2 green sectors, 2 blue sectors, and 6 yellow sectors. If the spinner lands on a red sector, the player wins 2 tokens. If the spinner lands on a green sector, the player wins 2 tokens. If the spinner lands on a blue sector, the player wins 2 tokens. If the spinner lands on a yellow sector, the player loses 3 tokens.

Question

At a County fair there is a spinner game with 12 sectors: 2 red sectors, 2 green sectors, 2 blue sectors, and 6 yellow sectors. If the spinner lands on a red sector, the player wins 2 tokens. If the spinner lands on a green sector, the player wins 2 tokens. If the spinner lands on a blue sector, the player wins 2 tokens. If the spinner lands on a yellow sector, the player loses 3 tokens.

Is this game fair for the player and how much will the player win or lose on an average over time? Explain.

Answer 1

To determine if the game is fair for the player, we need to calculate the average number of tokens won or lost per spin.

First, let's calculate the probability of landing on each sector:
- Probability of landing on a red sector: 2/12 = 1/6
- Probability of landing on a green sector: 2/12 = 1/6
- Probability of landing on a blue sector: 2/12 = 1/6
- Probability of landing on a yellow sector: 6/12 = 1/2

Now, let's calculate the expected outcome for each sector:
- For red, green, and blue sectors: Probability of winning 2 tokens = 1/6 * 2 = 1/3 tokens
- For yellow sectors: Probability of losing 3 tokens = 1/2 * -3 = -3/2 tokens (negative because it's a loss)

To calculate the overall expected outcome, we sum up the expected outcomes of all sectors:
Expected outcome = 1/3 + 1/3 + 1/3 - 3/2 = 2/3 - 3/2 = 4/6 - 9/6 = -5/6 tokens

Therefore, over time, the player can expect to lose an average of 5/6 tokens per spin. Since the player loses tokens on average, the game is not fair and is not favorable for the player.