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
StartFraction 3 Over 113 EndFraction
.
(Type an integer or a simplified fraction.)
Part 2
The theoretical 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
StartFraction 3 Over 113 EndFraction
.
(Type an integer or a simplified fraction.)
Part 2
The theoretical probability of choosing the name Grace is
enter your response here.
(Type an integer or a simplified fraction.)
Answers
Answered by
GPT-4o mini
### Part 1: Experimental Probability
The experimental probability of an event is calculated by taking the number of times the event occurs divided by the total number of trials. In this case, the name Grace was chosen 3 times out of 113 total choices.
The formula for experimental probability is:
\[
P(\text{Grace}) = \frac{\text{Number of times Grace is chosen}}{\text{Total number of choices}} = \frac{3}{113}
\]
So, the experimental probability of choosing the name Grace is:
\[
\frac{3}{113}
\]
---
### Part 2: Theoretical Probability
The theoretical probability is based on the assumption that each name has an equal chance of being chosen. Since there are 9 different names in the hat, the probability of choosing any specific name (including Grace) can be calculated as follows:
\[
P(\text{Grace}) = \frac{1}{\text{Total number of names}} = \frac{1}{9}
\]
So, the theoretical probability of choosing the name Grace is:
\[
\frac{1}{9}
\]
---
### Discussion on Changes in Number of Names
1. **If the Number of Names Increases**:
- Suppose the number of names in the hat increases to 12. The theoretical probability would change to \( \frac{1}{12} \). This would decrease the probability of selecting Grace, as there are now more names for the selection.
2. **If the Number of Names Decreases**:
- If the number of names decreases to 5, the theoretical probability would change to \( \frac{1}{5} \). This increases the probability of selecting Grace, as there are fewer names in the pool to choose from.
3. **Experimental Probability**:
- The experimental probability can vary based on the actual outcomes of each trial (i.e., how many times Grace is chosen). The more choices made, the closer the experimental probability is often expected to get to the theoretical probability, especially if the trials are far from being independent or if the sampling method changes.
In summary, the experimental probability reflects the results of real trials, while the theoretical probability reflects the outcome we would expect based on equal likelihood among all options, dependent on the total number of options available.
The experimental probability of an event is calculated by taking the number of times the event occurs divided by the total number of trials. In this case, the name Grace was chosen 3 times out of 113 total choices.
The formula for experimental probability is:
\[
P(\text{Grace}) = \frac{\text{Number of times Grace is chosen}}{\text{Total number of choices}} = \frac{3}{113}
\]
So, the experimental probability of choosing the name Grace is:
\[
\frac{3}{113}
\]
---
### Part 2: Theoretical Probability
The theoretical probability is based on the assumption that each name has an equal chance of being chosen. Since there are 9 different names in the hat, the probability of choosing any specific name (including Grace) can be calculated as follows:
\[
P(\text{Grace}) = \frac{1}{\text{Total number of names}} = \frac{1}{9}
\]
So, the theoretical probability of choosing the name Grace is:
\[
\frac{1}{9}
\]
---
### Discussion on Changes in Number of Names
1. **If the Number of Names Increases**:
- Suppose the number of names in the hat increases to 12. The theoretical probability would change to \( \frac{1}{12} \). This would decrease the probability of selecting Grace, as there are now more names for the selection.
2. **If the Number of Names Decreases**:
- If the number of names decreases to 5, the theoretical probability would change to \( \frac{1}{5} \). This increases the probability of selecting Grace, as there are fewer names in the pool to choose from.
3. **Experimental Probability**:
- The experimental probability can vary based on the actual outcomes of each trial (i.e., how many times Grace is chosen). The more choices made, the closer the experimental probability is often expected to get to the theoretical probability, especially if the trials are far from being independent or if the sampling method changes.
In summary, the experimental probability reflects the results of real trials, while the theoretical probability reflects the outcome we would expect based on equal likelihood among all options, dependent on the total number of options available.