Let's go through the questions one by one, breaking them down into detailed steps.
1. How many of the outcomes are favorable?
- From the setup of the problem, you're asked to click Clear and then run 10 trials 10 more times, resulting in a total of 100 trials.
- Since the final result given in the solution is `None`, it indicates that there are no favorable outcomes observed. Therefore, the number of favorable outcomes is 0.
- So, you would fill in:
```
0
```
2. What is the probability of landing on a 1 now?
- The number of favorable outcomes (landing on a 1) is 0.
- The total number of trials is 100.
- Probability is calculated as the number of favorable outcomes divided by the total number of trials.
- So, the probability of landing on a 1 is:
[tex]\[
\frac{0}{100} = \frac{0}{1} = 0
\][/tex]
- You would fill in the fraction as:
```
0/100
```
3. Compare the experimental probability with the theoretical probability of [tex]\( P(x=1) = \frac{1}{3} \)[/tex]. Do you think an experiment with 100 trials produces a probability that is more reliable than an experiment with 10 trials?
- The theoretical probability of landing on a 1 is given as [tex]\( \frac{1}{3} \)[/tex].
- The experimental probability from our 100 trials is 0, which is much lower than [tex]\( \frac{1}{3} \)[/tex].
- Generally, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability due to the Law of Large Numbers. This means that larger sample sizes tend to produce more reliable and accurate estimates of the true probability.
- Although our experimental probability of 0 is quite different from the theoretical probability of [tex]\( \frac{1}{3} \)[/tex], it's usually expected that a larger number of trials (like 100 trials) would give a result closer to the theoretical probability compared to a smaller number of trials (like 10 trials).
The completed responses to the questions would therefore be:
1. How many of the outcomes are favorable?
```
0
```
2. What is the probability of landing on a 1 now?
```
0/100
```
3. Compare the experimental probability with the theoretical probability of [tex]\( P(x=1) = \frac{1}{3} \)[/tex]. Do you think an experiment with 100 trials produces a probability that is more reliable than an experiment with 10 trials?
```
Yes, an experiment with 100 trials is generally more reliable than one with 10 trials as it is more likely to produce results closer to the theoretical probability.
```
