Answer :
Sure, let's go through each part of this problem step-by-step.
### (a) Experimental Probability of Getting an 8
To determine the experimental probability of getting an 8, we use the data provided. Specifically, we need the number of times the outcome was 8 and the total number of trials.
From the given data:
- Number of times the outcome was 8: [tex]\( 12 \)[/tex]
- Total number of trials: [tex]\( 100 \)[/tex]
The experimental probability [tex]\( P_{\text{exp}}(8) \)[/tex] is given by the formula:
[tex]\[ P_{\text{exp}}(8) = \frac{\text{Number of times 8 occurred}}{\text{Total number of trials}} \][/tex]
Substituting in the values:
[tex]\[ P_{\text{exp}}(8) = \frac{12}{100} = 0.12 \][/tex]
### (b) Theoretical Probability of Getting an 8
Assuming that the machine is fair, each digit from 0 to 9 has an equal chance of being selected. Since there are 10 possible outcomes and all are equally likely, the theoretical probability [tex]\( P_{\text{theo}}(8) \)[/tex] of getting an 8 is:
[tex]\[ P_{\text{theo}}(8) = \frac{1}{10} = 0.1 \][/tex]
### (c) Relationship Between Number of Trials and Experimental Probability
Here we are asked to determine which of the following statements is true:
- The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
- The smaller the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
- The experimental probability will never be very close to the theoretical probability, no matter the number of trials.
The correct statement is:
- The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
This is because, by the Law of Large Numbers, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.
### Summary
So, the answers to the parts of the question are:
(a) The experimental probability of getting an 8 is [tex]\( \boxed{0.12} \)[/tex].
(b) The theoretical probability of getting an 8 is [tex]\( \boxed{0.1} \)[/tex].
(c) The correct statement is [tex]\( \boxed{1} \)[/tex], i.e., "The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability."
### (a) Experimental Probability of Getting an 8
To determine the experimental probability of getting an 8, we use the data provided. Specifically, we need the number of times the outcome was 8 and the total number of trials.
From the given data:
- Number of times the outcome was 8: [tex]\( 12 \)[/tex]
- Total number of trials: [tex]\( 100 \)[/tex]
The experimental probability [tex]\( P_{\text{exp}}(8) \)[/tex] is given by the formula:
[tex]\[ P_{\text{exp}}(8) = \frac{\text{Number of times 8 occurred}}{\text{Total number of trials}} \][/tex]
Substituting in the values:
[tex]\[ P_{\text{exp}}(8) = \frac{12}{100} = 0.12 \][/tex]
### (b) Theoretical Probability of Getting an 8
Assuming that the machine is fair, each digit from 0 to 9 has an equal chance of being selected. Since there are 10 possible outcomes and all are equally likely, the theoretical probability [tex]\( P_{\text{theo}}(8) \)[/tex] of getting an 8 is:
[tex]\[ P_{\text{theo}}(8) = \frac{1}{10} = 0.1 \][/tex]
### (c) Relationship Between Number of Trials and Experimental Probability
Here we are asked to determine which of the following statements is true:
- The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
- The smaller the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
- The experimental probability will never be very close to the theoretical probability, no matter the number of trials.
The correct statement is:
- The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
This is because, by the Law of Large Numbers, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.
### Summary
So, the answers to the parts of the question are:
(a) The experimental probability of getting an 8 is [tex]\( \boxed{0.12} \)[/tex].
(b) The theoretical probability of getting an 8 is [tex]\( \boxed{0.1} \)[/tex].
(c) The correct statement is [tex]\( \boxed{1} \)[/tex], i.e., "The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability."
