To analyze Karen's spinner results and compute the necessary probabilities, let's complete each part step-by-step.
(a) Experimental Probability of Landing on Black:
To find the experimental probability, we use the formula:
\[ P(\text{Black}) = \frac{\text{Number of Spins on Black}}{\text{Total Spins}}. \]
From the data provided:
- Number of spins on Black = 206
- Total spins = 1000
Now we calculate:
\[ P(\text{Black}) = \frac{206}{1000} = 0.206. \]
Rounding to the nearest thousandth, we get:
\[ P(\text{Black}) \approx 0.206. \]
(b) Theoretical Probability of Landing on Black:
Assuming the spinner is fair and has 5 equally sized slices (White, Grey, Black, and two other colors), the theoretical probability of landing on any specific color (including Black) is given by:
\[ P(\text{Black}) = \frac{1}{\text{Total number of slices}} = \frac{1}{5} = 0.2. \]
(c) True Statement Regarding Large Number of Spins:
The correct statement regarding the experimental vs. theoretical probabilities with a large number of spins is:
With a large number of spins, there might be a difference between the experimental and theoretical probabilities, but the difference should be small.
This acknowledges that while the experimental probability may vary due to random chance, over many trials, the experimental probability is expected to approximate the theoretical probability closely.
Summary of Results:
(a) Experimental Probability of landing on Black: 0.206
(b) Theoretical Probability of landing on Black: 0.2
(c) True Statement: With a large number of spins, there might be a difference between the experimental and theoretical probabilities, but the difference should be small.