Question

roll the number cube to find the observed frequency. dice image table with columns: number (1, 2, 3, 4, 5, 6, total); observed frequency (0, 0, 0, 0, 0, 0, ) roll one time. roll ten times. which statements are true? check all that apply. □ as the number of trials increases, experimental probability is closer to theoretical probability. □ as the number of trials increases, theoretical probability changes to more closely match the experimental probability. □ as the number of trials increases, there is no change in the experimental probabilities. □ as the number of trials increases, there is no change in the theoretical probabilities.

Explanation:

• For the relationship between experimental and theoretical probability: According to the law of large numbers, as the number of trials (n) in a random experiment increases, the experimental probability ($P_{experimental}=\frac{\text{Number of successful trials}}{n}$) will approach the theoretical probability ($P_{theoretical}$, which is calculated based on the inherent nature of the event, e.g., for a fair die, $P_{theoretical}(rolling\ a\ 1)=\frac{1}{6}$). So "As the number of trials increases, experimental probability is closer to theoretical probability" is correct.
• Theoretical probability: It is determined by the sample space and the favorable outcomes of the event. For example, in the case of rolling a fair die, the theoretical probability of each face showing up is $\frac{1}{6}$ and it does not depend on the number of times we roll the die. So "As the number of trials increases, there is no change in the theoretical probabilities" is correct.
• Experimental probability: It is calculated based on the actual results of the trials. As the number of trials changes, the number of successful trials can change, so the experimental probability can also change. So "As the number of trials increases, there is no change in the experimental probabilities" is wrong.
• The theoretical probability is a fixed value for a given event and it does not adjust to match the experimental probability. So "As the number of trials increases, theoretical probability changes to more closely match the experimental probability" is wrong.

Answer:

• As the number of trials increases, experimental probability is closer to theoretical probability.
• As the number of trials increases, there is no change in the experimental probabilities. (This statement is incorrect. The correct idea is that experimental probability approaches theoretical probability as trials increase, and experimental probabilities can change with more trials, but the last two options: "As the number of trials increases, there is no change in the experimental probabilities" is wrong, "As the number of trials increases, there is no change in the theoretical probabilities" is correct. Wait, let's re - analyze:

Theoretical probability is based on the nature of the event (e.g., for a fair die, the probability of rolling a 1 is $\frac{1}{6}$ and it doesn't change with the number of trials). Experimental probability is the ratio of the number of times an event occurs to the number of trials. As the number of trials increases, experimental probability tends to get closer to theoretical probability. And theoretical probability is a fixed value for a given event, so it doesn't change with the number of trials. The experimental probability can change as the number of trials changes (because the number of times the event occurs can change).

So the correct statements are:

1. As the number of trials increases, experimental probability is closer to theoretical probability.

1. As the number of trials increases, there is no change in the theoretical probabilities.

(Assuming the options are:

Option 1: As the number of trials increases, experimental probability is closer to theoretical probability.

Option 2: As the number of trials increases, theoretical probability changes to more closely match the experimental probability. (Wrong)

Option 3: As the number of trials increases, there is no change in the experimental probabilities. (Wrong)

Option 4: As the number of trials increases, there is no change in the theoretical probabilities. (Correct)

So the correct answers are Option 1 and Option 4)