a The expected value of one spin is the sum of the products of the probability of spinning each sector and their corresponding earnings.
B
b Redo your calculations from Part A with respect to region A.
C
c Label region A's expected winnings e.
A
a 4
B
b 0
C
c - 16
Practice makes perfect
a To find the expected value for a spin, we have to calculate the expected value from each region which will depend on the probability of spinning that region and its earnings. From the diagram, we know that one region has a central angle of 180^(∘) and another has a central angle of 90^(∘).
Since the region with 0 and -8 has the same probability of happening, they must occupy the same area of the spinner. This must mean they have the same central angle, θ. With this information, we can write and solve an equation containing θ.
θ+θ+90^(∘)+180^(∘)=360^(∘) ⇔ θ =45^(∘)
Let's add this information to the diagram.
To find the expected value, we have to multiply the probability of spinning a certain field with what you win from that field. The probability of spinning a certain field is its central angle divided by 360^(∘).
Value
Probability
Value * Probability
Expected value
- 8
45^(∘)/360^(∘)
- 8(45^(∘)/360^(∘))
- 1
0
45^(∘)/360^(∘)
0(45^(∘)/360^(∘))
0
8
90^(∘)/360^(∘)
8(90^(∘)/360^(∘))
2
6
180^(∘)/360^(∘)
6(180^(∘)/360^(∘))
3
The expected value is the sum of the last column in the table above.
- 1+0+2+3=4
b The expected value from the three regions that are not A are unchanged. The only calculation we have to redo is region A's. Instead of earning 8, you earn - 4.
- 4(90^(∘)/360^(∘))=- 2
Finally, we will add the expected value from all four regions.
- 1+0+(- 2)+3=0
c Let's label region A's expected value e. Since the expected value of spinning the wheel once should be 0, we can write and solve the following equation.
- 1+0+e+3=0 ⇔ e=- 4The expected value from region A should be - 4. By equating this number with the expression describing the expected value from region A, we determine what this region must give.
A(90^(∘)/360^(∘))=- 4
Let's solve this equation for A.
