The spinner below shows equally sized slices. Karen spun the dial times and got the following results.

Question

Outcome White Grey Black
Number of Spins
Fill in the table below. Round your answers to the nearest thousandth.
(a) Assuming that the spinner is fair, compute the theoretical probability of landing on black or grey.

(b) From Karen's results, compute the experimental probability of landing on black or grey.
(c) Assuming that the spinner is fair, choose the statement below that is true:

As the number of spins increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

The experimental and theoretical probabilities must always be equal.

As the number of spins increases, we expect the experimental and theoretical probabilities to become farther apart.

Answer 1

To complete your request, I need some additional information from you since the specific data (number of spins, the number of outcomes for each color) hasn't been provided. However, I can guide you through how to find the answers based on hypothetical values, and you can adjust them with the actual numbers as per your results.

Let's break down the steps together:

(a) Theoretical Probability:

For a fair spinner with equally sized slices for White, Grey, and Black, let's assume there are 3 colors. The theoretical probability of landing on either black or grey can be computed as follows:

• Each color (White, Grey, Black) has a probability of \(\frac{1}{3}\).
• Therefore, the probability of landing on Grey or Black combined: \[ P(\text{black or grey}) = P(\text{black}) + P(\text{grey}) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \approx 0.667 \] You would round 0.667 to the nearest thousandth as necessary.

(b) Experimental Probability:

1. Count the number of spins resulting in Black and Grey outcomes from Karen’s actual results.

   Let’s say Karen spun the spinner \(N\) times, with results like this:

   • Number of Blacks = \(B\)
   • Number of Greys = \(G\)

2. The experimental probability can be calculated as: \[ P(\text{black or grey}) = \frac{\text{Number of blacks} + \text{Number of greys}}{\text{Total number of spins}} = \frac{B + G}{N} \] Make sure to round this decimal to the nearest thousandth.

(c) Statement Evaluation:

Given the nature of probability, the correct statement is typically:

• As the number of spins increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

This is due to the Law of Large Numbers, which states that as an experiment is conducted more times, the experimental probability will tend to reflect the theoretical probability more accurately.

Example:

Let’s illustrate with hypothetical values:

• Assume Karen spun the spinner 100 times, resulting in:
  • White: 40 times
  • Grey: 30 times
  • Black: 30 times

So:

• Experimental probability of Black or Grey: \[ \frac{30 + 30}{100} = \frac{60}{100} = 0.600 \]
• Theoretical probability (calculated previously) = 0.667

You may use the approach above with your actual data to fill out the table.