Answer :
Answer:
[tex]E(X) = -1.5[/tex]
No, the game is not fair since the expected value represents a loss.
Step-by-step explanation:
The expected value of this game is calculated using
[tex]E(X) = \sum(x_{i}p_{i})[/tex]
Where [tex]x_{i}[/tex] is the net gain of each outcome and [tex]p_{i}[/tex] is the probability of each outcome.
When a die is rolled, the probability of getting each outcome is
[tex]p = \frac{1}{6}[/tex]
Where 6 is the total number of possible outcomes.
The cost of playing the game is $5
The net gain for each outcome is given by
[tex]x_{1} = 1 - 5 =-4\\x_{2} = 2 - 5 =-3\\x_{3} = 3 - 5 =-2\\x_{4} = 4 - 5 =-1\\x_{5} = 5 - 5 = 0\\x_{6} = 6 - 5 =1\\[/tex]
Now we can find the expected value of this game,[tex]E(X) = (x_{1} \cdot p_{1}) + (x_{2} \cdot p_{2}) +(x_{3} \cdot p_{3})+(x_{4} \cdot p_{4})+(x_{5} \cdot p_{5}) + (x_{6} \cdot p_{6})[/tex]
Since the probability of each outcome is same
[tex]E(X) = p(x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{5}+x_{6}) \\E(X) = \frac{1}{6} (-4 -3 -2 -1+0+1)\\E(X) = \frac{1}{6} (-9)\\E(X) = \frac{-9}{6} \\E(X) = -1.5[/tex]
Therefore, we can conclude that this game is not fair since the expected value is negative which represents a loss rather than a gain.