Sure, let's solve the problem step by step.
### a. Filling out the Missing Values in the Table
First, we need to determine the probabilities for each colored sector.
- The wheel has 7 sectors in total.
- There are 2 red sectors out of 7, so the probability of landing on a red sector (P(Red)) is [tex]\( \frac{2}{7} \)[/tex].
- There are 2 purple sectors out of 7, so the probability of landing on a purple sector (P(Purple)) is [tex]\( \frac{2}{7} \)[/tex].
- There are 2 yellow sectors out of 7, so the probability of landing on a yellow sector (P(Yellow)) is [tex]\( \frac{2}{7} \)[/tex].
- There is 1 blue sector out of 7, so the probability of landing on a blue sector (P(Blue)) is [tex]\( \frac{1}{7} \)[/tex].
Now we create the table with the points for each color and their corresponding probabilities:
| Color | Points | Probability |
|--------|--------|--------------------|
| Red | -1 | [tex]\( \frac{2}{7} \)[/tex] ≈ 0.2857142857142857 |
| Purple | 0 | [tex]\( \frac{2}{7} \)[/tex] ≈ 0.2857142857142857 |
| Yellow | 1 | [tex]\( \frac{2}{7} \)[/tex] ≈ 0.2857142857142857 |
| Blue | 3 | [tex]\( \frac{1}{7} \)[/tex] ≈ 0.14285714285714285 |
### b. Calculating the Expected Value
To find the expected value of the points after one spin, we use the formula for the expected value of a discrete random variable:
[tex]\[ E(X) = \sum (x_i \cdot P(x_i)) \][/tex]
where [tex]\( x_i \)[/tex] is the number of points for each sector, and [tex]\( P(x_i) \)[/tex] is the probability of landing on that sector.
Let's break it down:
- For Red: Points = -1, Probability = [tex]\( \frac{2}{7} \)[/tex]
- For Purple: Points = 0, Probability = [tex]\( \frac{2}{7} \)[/tex]
- For Yellow: Points = 1, Probability = [tex]\( \frac{2}{7} \)[/tex]
- For Blue: Points = 3, Probability = [tex]\( \frac{1}{7} \)[/tex]
Now, we calculate the expected value:
[tex]\[ E(X) = (-1 \cdot \frac{2}{7}) + (0 \cdot \frac{2}{7}) + (1 \cdot \frac{2}{7}) + (3 \cdot \frac{1}{7}) \][/tex]
Calculating each term separately:
[tex]\[ -1 \cdot \frac{2}{7} = -\frac{2}{7} \][/tex]
[tex]\[ 0 \cdot \frac{2}{7} = 0 \][/tex]
[tex]\[ 1 \cdot \frac{2}{7} = \frac{2}{7} \][/tex]
[tex]\[ 3 \cdot \frac{1}{7} = \frac{3}{7} \][/tex]
Adding these together:
[tex]\[ E(X) = -\frac{2}{7} + 0 + \frac{2}{7} + \frac{3}{7} = \frac{3}{7} \][/tex]
Therefore, the expected value for one spin is:
[tex]\[ E(X) = \frac{3}{7} ≈ 0.42857142857142855 \][/tex]
### Summary
- a. The table with the missing values:
| Color | Points | Probability |
|--------|--------|--------------------|
| Red | -1 | [tex]\( \frac{2}{7} \)[/tex] ≈ 0.2857142857142857 |
| Purple | 0 | [tex]\( \frac{2}{7} \)[/tex] ≈ 0.2857142857142857 |
| Yellow | 1 | [tex]\( \frac{2}{7} \)[/tex] ≈ 0.2857142857142857 |
| Blue | 3 | [tex]\( \frac{1}{7} \)[/tex] ≈ 0.14285714285714285 |
- b. The expected value after one spin is:
[tex]\[ E(X) ≈ 0.42857142857142855 \][/tex]
