Let's analyze the given scenario step by step to determine which color's experimental probability is closest to the theoretical probability.
### Step 1: Calculating Experimental Probabilities
First, we need to find the experimental probability for each color. The experimental probability [tex]\( P(E) \)[/tex] is calculated as:
[tex]\[ P(E) = \frac{\text{Frequency of the color}}{\text{Total number of spins}} \][/tex]
Given the total number of spins is 625, we now compute the experimental probabilities for each color:
- Orange:
[tex]\[ P(Orange) = \frac{118}{625} \approx 0.1888 \][/tex]
- Purple:
[tex]\[ P(Purple) = \frac{137}{625} \approx 0.2192 \][/tex]
- Brown:
[tex]\[ P(Brown) = \frac{122}{625} \approx 0.1952 \][/tex]
- Yellow:
[tex]\[ P(Yellow) = \frac{106}{625} \approx 0.1696 \][/tex]
- Green:
[tex]\[ P(Green) = \frac{142}{625} \approx 0.2272 \][/tex]
### Step 2: Theoretical Probability
Since the spinner has five equal-sized sections, the theoretical probability [tex]\( P(T) \)[/tex] for each color is:
[tex]\[ P(T) = \frac{1}{5} = 0.2 \][/tex]
### Step 3: Comparing Experimental Probability and Theoretical Probability
We now need to calculate the absolute difference between the experimental and theoretical probabilities for each color:
- Orange:
[tex]\[ |P(Orange) - P(T)| = |0.1888 - 0.2| = 0.0112 \][/tex]
- Purple:
[tex]\[ |P(Purple) - P(T)| = |0.2192 - 0.2| = 0.0192 \][/tex]
- Brown:
[tex]\[ |P(Brown) - P(T)| = |0.1952 - 0.2| = 0.0048 \][/tex]
- Yellow:
[tex]\[ |P(Yellow) - P(T)| = |0.1696 - 0.2| = 0.0304 \][/tex]
- Green:
[tex]\[ |P(Green) - P(T)| = |0.2272 - 0.2| = 0.0272 \][/tex]
### Step 4: Determining the Closest Color
The color with the smallest absolute difference between its experimental probability and the theoretical probability is:
- Orange: 0.0112
- Purple: 0.0192
- Brown: 0.0048
- Yellow: 0.0304
- Green: 0.0272
The smallest difference is 0.0048 for Brown.
### Conclusion
The experimental probability for the color Brown is the closest to the theoretical probability. The absolute difference is only 0.0048, which is the smallest among all the colors.
