Answer :
Sure, let's break down this problem step by step:
### Part (a): Compute the Experimental Probability of Getting a 5
The experimental probability is calculated based on the results from the given trials. We are to determine how many times the outcome was 5 and divide it by the total number of trials.
1. Number of Trials for Outcome 5: From the table, outcome 5 occurred 6 times.
2. Total Number of Trials: There were 50 trials in total.
The formula for the experimental probability is:
[tex]\[ \text{Experimental Probability} = \frac{\text{Number of Trials Outcome 5}}{\text{Total Number of Trials}} \][/tex]
Substituting the values:
[tex]\[ \text{Experimental Probability} = \frac{6}{50} = 0.12 \][/tex]
Therefore, the experimental probability of getting a 5 is:
[tex]\[ \boxed{0.12} \][/tex]
### Part (b): Compute the Theoretical Probability of Getting a 5
Assuming the machine is fair, each outcome (from 0 to 9) is equally likely. There are 10 possible outcomes in total.
The formula for the theoretical probability of any single outcome in a fair situation is:
[tex]\[ \text{Theoretical Probability} = \frac{1}{\text{Number of Possible Outcomes}} \][/tex]
In this case:
[tex]\[ \text{Theoretical Probability} = \frac{1}{10} = 0.1 \][/tex]
Therefore, the theoretical probability of getting a 5 is:
[tex]\[ \boxed{0.1} \][/tex]
### Part (c): Choose the True Statement
The three statements provided are:
1. As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.
2. The experimental and theoretical probabilities must always be equal.
3. As the number of trials increases, we expect the experimental and theoretical probabilities to become farther apart.
The correct statement is:
1. As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.
This statement is true according to the Law of Large Numbers, which states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. Therefore, the correct statement is:
[tex]\[ \boxed{1} \][/tex]
To summarize, the solutions are:
(a) Experimental Probability: \( \boxed{0.12} \)
(b) Theoretical Probability: \( \boxed{0.1} \)
(c) True Statement: [tex]\( \boxed{1} \)[/tex]
### Part (a): Compute the Experimental Probability of Getting a 5
The experimental probability is calculated based on the results from the given trials. We are to determine how many times the outcome was 5 and divide it by the total number of trials.
1. Number of Trials for Outcome 5: From the table, outcome 5 occurred 6 times.
2. Total Number of Trials: There were 50 trials in total.
The formula for the experimental probability is:
[tex]\[ \text{Experimental Probability} = \frac{\text{Number of Trials Outcome 5}}{\text{Total Number of Trials}} \][/tex]
Substituting the values:
[tex]\[ \text{Experimental Probability} = \frac{6}{50} = 0.12 \][/tex]
Therefore, the experimental probability of getting a 5 is:
[tex]\[ \boxed{0.12} \][/tex]
### Part (b): Compute the Theoretical Probability of Getting a 5
Assuming the machine is fair, each outcome (from 0 to 9) is equally likely. There are 10 possible outcomes in total.
The formula for the theoretical probability of any single outcome in a fair situation is:
[tex]\[ \text{Theoretical Probability} = \frac{1}{\text{Number of Possible Outcomes}} \][/tex]
In this case:
[tex]\[ \text{Theoretical Probability} = \frac{1}{10} = 0.1 \][/tex]
Therefore, the theoretical probability of getting a 5 is:
[tex]\[ \boxed{0.1} \][/tex]
### Part (c): Choose the True Statement
The three statements provided are:
1. As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.
2. The experimental and theoretical probabilities must always be equal.
3. As the number of trials increases, we expect the experimental and theoretical probabilities to become farther apart.
The correct statement is:
1. As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.
This statement is true according to the Law of Large Numbers, which states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. Therefore, the correct statement is:
[tex]\[ \boxed{1} \][/tex]
To summarize, the solutions are:
(a) Experimental Probability: \( \boxed{0.12} \)
(b) Theoretical Probability: \( \boxed{0.1} \)
(c) True Statement: [tex]\( \boxed{1} \)[/tex]
