Answer:
Option D- The expected value is -0.5, and the game is not fair. So the player will lose about 0.5 tokens for a single spin over time.
Step-by-step explanation:
Given : 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.
A green sector- the player wins 2 tokens.
A blue sector- the player wins 2 tokens.
A yellow sector- the player loses 3 tokens.
To find : Is this game fair for the player and how much will the player win or lose on an average over time?
Solution :
First we find the probability of each sector,
[tex]\text{Probability}=\frac{\text{Favorable outcome}}{\text{Total number of outcome}}[/tex]
Total outcome = 12
1) [tex]\text{Probability(on red)}=\frac{2}{12}=\frac{1}{6}[/tex]
2) [tex]\text{Probability(on green)}=\frac{2}{12}=\frac{1}{6}[/tex]
3) [tex]\text{Probability(on blue)}=\frac{2}{12}=\frac{1}{6}[/tex]
4) [tex]\text{Probability(on yellow)}=\frac{6}{12}=\frac{1}{2}[/tex]
Now, For a win the profit is w=2 tokens.
For a loss, the profit is l= -3 tokens.
Now, The expected value is the product of probability and its profit/loss.
[tex]E(x)=P(R)\times w+P(G)\times w+P(B)\times w+P(Y)\times l[/tex]
[tex]E(x)=\frac{1}{6}\times 2+\frac{1}{6}\times 2+\frac{1}{6}\times 2+\frac{1}{2}\times (-3)[/tex]
[tex]E(x)=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}-\frac{3}{2}[/tex]
[tex]E(x)=1-\frac{3}{2}[/tex]
[tex]E(x)=-\frac{1}{2}[/tex]
[tex]E(x)=-0.5[/tex]
Therefore, The game is not fair. So the player will lose about 0.5 tokens for a single spin over time.
Hence, Option D is correct.
