Alyssa and Gabriel play the following game. Alyssa rolls a die. If she rolls a 1, 2, or 3, Gabriel gives Alyssa $4. If Alyssa rolls a 4 or 5, Gabriel gives Alyssa $1. However, if Alyssa rolls a 6, she gives Gabriel $12.
a) Determine Alyssa's expectation.
b) Determine Gabriel's expectation.
11.3 #12

The expected value is the probability of winning multiplied by the value you get when you win, plus the probability of losing multiplied by the value you get when you lose (which is negative as it is a loss).

Part a) Determine Alyssa's expectation

we know that

1) If Alyssa rolls a 1, 2, or 3, Gabriel gives Alyssa $4

The probability is 3/6

so

we have

[tex]\frac{3}{6}(4)=\$2[/tex]

2) If Alyssa rolls a 4, or 5, Gabriel gives Alyssa $1

The probability is 2/6

so

we have

[tex]\frac{2}{6}(1)=\$0.33[/tex]

3) if Alyssa rolls a 6, she gives Gabriel $12

The probability is 1/6

so

we have

[tex]-\frac{1}{6}(12)=-\$2[/tex] ---> is negative because is a loss

therefore

Alyssa's expectation is equal to

[tex]\$2+\$0.33-\$2=\$0.33[/tex]

Part b) Determine Gabriel's expectation

we know that

1) If Alyssa rolls a 1, 2, or 3, Gabriel gives Alyssa $4

Is a loss for Gabriel

The probability is 3/6

so

we have

[tex]-\frac{3}{6}(4)=-\$2[/tex] ---> is negative because is a loss

2) If Alyssa rolls a 4, or 5, Gabriel gives Alyssa $1

Is a loss for Gabriel

The probability is 2/6

so

we have

[tex]-\frac{2}{6}(1)=-\$0.33[/tex] ---> is negative because is a loss

3) if Alyssa rolls a 6, she gives Gabriel $12

Is a win for Gabriel

The probability is 1/6

so

we have

[tex]\frac{1}{6}(12)=\$2[/tex]

therefore

Gabriel's expectation is equal to

[tex]-\$2-\$0.33+\$2=-\$0.33[/tex]