Question
Challenge Nine different names were put into a hat. A name is chosen 113 times and the name Grace is chosen 3 times. What is the experimental probability of the name Grace being chosen? What is the theoretical probability of the name Grace being chosen? Use pencil and paper. Explain how each probability would change if the number of names in the hat were different.
Question content area bottom
Part 1
The experimental probability of choosing the name Grace is
enter your response here.
(Type an integer or a simplified fraction.)
Question content area bottom
Part 1
The experimental probability of choosing the name Grace is
enter your response here.
(Type an integer or a simplified fraction.)
Answers
Answered by
GPT-4o mini
To determine the probabilities, we start with the definitions:
1. **Experimental Probability**: This is calculated based on the outcomes of an experiment. It is given by the formula:
\[
P(E) = \frac{\text{Number of times the event occurs}}{\text{Total number of trials}}
\]
In this case, the event is choosing the name Grace.
From the data provided:
- Number of times Grace was chosen = 3
- Total number of times names were chosen = 113
Therefore, the experimental probability of choosing the name Grace is:
\[
P(\text{Grace}) = \frac{3}{113}
\]
2. **Theoretical Probability**: This is calculated based on the expected outcomes in an ideal situation. If there are 9 different names in the hat, and assuming each name has an equal chance of being selected, the theoretical probability of choosing Grace would be:
\[
P(\text{Grace}) = \frac{1}{\text{Total number of names}} = \frac{1}{9}
\]
Now, let's summarize:
- **Experimental Probability of Grace being chosen**: \( \frac{3}{113} \)
- **Theoretical Probability of Grace being chosen**: \( \frac{1}{9} \)
### How Probabilities Change with a Different Number of Names
If the total number of names in the hat were different, the theoretical probability would change based on the new total:
- If there are **more names** (e.g., 10 names), the theoretical probability would then be \( \frac{1}{10} \).
- If there are **fewer names** (e.g., 5 names), the theoretical probability would be \( \frac{1}{5} \).
As for the experimental probability, it is dependent on the results of the specific trials (how many times Grace was chosen relative to the total number of choices). This value could still remain \( \frac{3}{113} \) unless the number of trials is changed or the outcomes of trials are different.
1. **Experimental Probability**: This is calculated based on the outcomes of an experiment. It is given by the formula:
\[
P(E) = \frac{\text{Number of times the event occurs}}{\text{Total number of trials}}
\]
In this case, the event is choosing the name Grace.
From the data provided:
- Number of times Grace was chosen = 3
- Total number of times names were chosen = 113
Therefore, the experimental probability of choosing the name Grace is:
\[
P(\text{Grace}) = \frac{3}{113}
\]
2. **Theoretical Probability**: This is calculated based on the expected outcomes in an ideal situation. If there are 9 different names in the hat, and assuming each name has an equal chance of being selected, the theoretical probability of choosing Grace would be:
\[
P(\text{Grace}) = \frac{1}{\text{Total number of names}} = \frac{1}{9}
\]
Now, let's summarize:
- **Experimental Probability of Grace being chosen**: \( \frac{3}{113} \)
- **Theoretical Probability of Grace being chosen**: \( \frac{1}{9} \)
### How Probabilities Change with a Different Number of Names
If the total number of names in the hat were different, the theoretical probability would change based on the new total:
- If there are **more names** (e.g., 10 names), the theoretical probability would then be \( \frac{1}{10} \).
- If there are **fewer names** (e.g., 5 names), the theoretical probability would be \( \frac{1}{5} \).
As for the experimental probability, it is dependent on the results of the specific trials (how many times Grace was chosen relative to the total number of choices). This value could still remain \( \frac{3}{113} \) unless the number of trials is changed or the outcomes of trials are different.