Mai rolled a number cube times and got the following results.

Question

Outcome Rolled
Number of Rolls
Fill in the table below. Round your answers to the nearest thousandth.
(a) From Mai's results, compute the experimental probability of rolling a .

(b) Assuming that the cube is fair, compute the theoretical probability of rolling a .
(c) Assuming that the cube is fair, choose the statement below that is true:

The experimental and theoretical probabilities must always be equal.

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

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

Answer 1

To assist you with the probabilities, I'll need the specific outcome from the results of the rolls that you want to analyze, as well as the total number of rolls Mai performed. Since those details are missing, I'll provide a general guide on how to fill in the table and answer the questions, and you can apply the numbers from Mai's results.

Steps to calculate probabilities:

1. Experimental Probability (a)

   • The experimental probability of rolling a specific outcome (let's say a "1") is calculated using the formula: \[ P(\text{outcome}) = \frac{\text{Number of times the outcome was rolled}}{\text{Total number of rolls}} \]
   • For example, if Mai rolled a "1" a total of 15 times out of 100 rolls, the experimental probability of rolling a "1" would be: \[ P(1) = \frac{15}{100} = 0.15 \]
   • Round this value to the nearest thousandth if necessary.

2. Theoretical Probability (b)

   • The theoretical probability of rolling a number on a fair six-sided die is: \[ P(\text{number}) = \frac{1}{6} \]
   • This is because there are 6 possible outcomes (1 through 6) and each has an equal chance of occurring.

3. Statement Analysis (c)

   • True Statement Choice: The correct choice among the given options is:
     • As the number of rolls increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.
   • This statement reflects the law of large numbers, which states that as more trials are conducted, the experimental probability is expected to converge to the theoretical probability.

Example Completion

For a concrete example, assume the following hypothetical results:

• Total rolls: 100
• Outcome "1" rolled: 20 times

Filling in the table:

(a) Experimental Probability of rolling a "1": \[ P(1) = \frac{20}{100} = 0.200 \quad \text{(rounded to the nearest thousandth)} \]

(b) Theoretical Probability of rolling a "1": \[ P(1) = \frac{1}{6} \approx 0.167 \quad \text{(rounded to the nearest thousandth)} \]

(c) True Statement: We select:

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

Make sure to plug in the actual numbers from Mai's results to calculate the probabilities based on her outcomes.