Sure, let's break down each part of the problem step-by-step.
### Part (a): Finding the Expected Value
To find the expected value [tex]\( E(X) \)[/tex] of the game, we use the formula for expected value:
[tex]\[ E(X) = \sum \left( x_i \cdot P(x_i) \right) \][/tex]
Here, [tex]\( x_i \)[/tex] represents the values on the chips, and [tex]\( P(x_i) \)[/tex] represents the probabilities of each value occurring. From the table, we have:
- For [tex]\( x = 2 \)[/tex], [tex]\( P(x) = 0.5 \)[/tex]
- For [tex]\( x = -1 \)[/tex], [tex]\( P(x) = 0.5 \)[/tex]
So, we calculate the expected value:
[tex]\[ E(X) = (2 \cdot 0.5) + (-1 \cdot 0.5) \][/tex]
Breaking it down:
[tex]\[ E(X) = (2 \cdot 0.5) + (-1 \cdot 0.5) = 1 - 0.5 = 0.5 \][/tex]
Thus, Miguel's expected value from playing the game is \[tex]$0.5.
### Part (b): Expected Winnings or Losses
Based on the expected value calculated in part (a), Miguel can expect to win \$[/tex]0.5 each time he plays the game. This is because the expected value tells us the average amount he can expect to win or lose per play, and in this case, it's a positive value, indicating a gain.
### Part (c): Making the Game Fair
To make the game fair, we need the expected value to be zero. Currently, the expected value is 0.5, indicating a gain for Miguel. We need to adjust one of the values to ensure that the average outcome is zero. Let [tex]\( x \)[/tex] be the new value that we need to assign to make the game fair.
The equation for the expected value becomes:
[tex]\[ (2 \cdot 0.5) + (-1 \cdot 0.5) + (x \cdot 0.5) = 0 \][/tex]
Simplifying this equation:
[tex]\[ 1 - 0.5 + 0.5x = 0 \][/tex]
[tex]\[ 0.5 + 0.5x = 0 \][/tex]
Subtract 0.5 from both sides:
[tex]\[ 0.5x = -0.5 \][/tex]
Divide by 0.5:
[tex]\[ x = -1 \][/tex]
So, to make the game fair—where the expected value is zero—the new value to assign should be [tex]\( -2 \)[/tex].
### Summary:
- b. Miguel's expected value from playing the game is \[tex]$0.5.
- c. Based on this expected value, Miguel can expect to win \$[/tex]0.5 each time he plays.
- d. To make the game fair, a value of [tex]\(-2\)[/tex] should be assigned to make the expected value zero.
This ensures that, on average, Miguel neither wins nor loses money.
