Question

At a fair, there is a spinner on a stall. The spinner has 4 equal sectors: 2 black sectors, 1 white sector, and 1 pink sector. If the spinner lands on a black sector, then there is a profit of $3. If the spinner lands on a white sector then then there is a profit of $2. But, if the spinner lands on any other sector, there is a loss of $1. Is this game fair?The game is not fair, and not favorable to the player.The game is not fair, but favorable to the player.The game is fair.The game is fair, and favorable to the player.

Answer 1

We have a game and it has an expected value for the prize.

Fair games have a expected value of 0, meaning that there is no expected loss or gain from it for the player.

We then can calculate the expected value for this game.

We list all the outcomes and their probabilities:

• If if lands on the black sector, the profit is $3. It can happen with a probability of 2/4 = 0.5.

,

• If it lands on the white sector, the profit is $2. It can happen with a probbility of 1/4 = 0.25.

,

• If it lands in the pink sector, the loss is $1. It can happen with a probbility of 1/4 = 0.25.

The expected value is the sum of all the possible outcomes weighted by their probability of happening.

Then, for this game, we will have:

`\begin{gathered} E(x)=\sum ^n_(i\mathop=i)p_ix_i \\ E(x)=0.5\cdot3+0.25\cdot2+0.25\cdot(-1) \\ E(x)=1.5+0.5-0.25 \\ E(x)=1.75 \end{gathered}`

The game has an expected profit of $1.75. This means that it is not a fair game and it is favorable to the player.

We can see from the outputs that are mostly positive (profit for the player) and with a high probability too.

Answer: The game is not fair, but favorable to the player.