Question
Challenge Six different names were put into a hat. A name is chosen 106 times and the name Josh is chosen 17 times. What is the experimental probability of the name Josh being chosen? What is the theoretical probability of the name Josh 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 Josh 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 Josh is
enter your response here.
(Type an integer or a simplified fraction.)
Answers
GPT-4o mini
To find the experimental probability of choosing the name Josh, we use the formula:
\[
\text{Experimental Probability} = \frac{\text{Number of times the event occurs}}{\text{Total number of trials}}
\]
In this case:
- The number of times the name Josh is chosen = 17
- The total number of times names are chosen = 106
So the experimental probability of choosing Josh is:
\[
\text{Experimental Probability} = \frac{17}{106}
\]
Next, we can express this fraction in simplified form. However, since 17 is a prime number and does not divide 106, it remains as is. Thus:
\[
\text{Experimental Probability} = \frac{17}{106}
\]
Now, for the theoretical probability, we assume that each name has an equal chance of being chosen. There are 6 different names in the hat, so the probability of choosing any one particular name, including Josh, is:
\[
\text{Theoretical Probability} = \frac{1}{\text{Total number of names}} = \frac{1}{6}
\]
**Summary:**
- The experimental probability of choosing the name Josh is \(\frac{17}{106}\).
- The theoretical probability of choosing the name Josh is \(\frac{1}{6}\).
### How These Probabilities Change with Different Number of Names
1. If the number of names in the hat decreases:
- The theoretical probability of any one name, including Josh, would increase. For example, if there were only 3 names, the theoretical probability would be \(\frac{1}{3}\).
- The experimental probability may still fluctuate based on how many times each name is randomly selected in actual trials.
2. If the number of names in the hat increases:
- The theoretical probability of any one name would decrease. For example, if there were 12 names, the theoretical probability would be \(\frac{1}{12}\).
- The experimental probability could change as well based on the outcomes during the trials—Josh might not be selected as much or could be selected more frequently, depending on chance over many trials.
In general, the experimental probability can vary greatly depending on the results of the specific trials, while the theoretical probability is fixed based on the number of names.
\[
\text{Experimental Probability} = \frac{\text{Number of times the event occurs}}{\text{Total number of trials}}
\]
In this case:
- The number of times the name Josh is chosen = 17
- The total number of times names are chosen = 106
So the experimental probability of choosing Josh is:
\[
\text{Experimental Probability} = \frac{17}{106}
\]
Next, we can express this fraction in simplified form. However, since 17 is a prime number and does not divide 106, it remains as is. Thus:
\[
\text{Experimental Probability} = \frac{17}{106}
\]
Now, for the theoretical probability, we assume that each name has an equal chance of being chosen. There are 6 different names in the hat, so the probability of choosing any one particular name, including Josh, is:
\[
\text{Theoretical Probability} = \frac{1}{\text{Total number of names}} = \frac{1}{6}
\]
**Summary:**
- The experimental probability of choosing the name Josh is \(\frac{17}{106}\).
- The theoretical probability of choosing the name Josh is \(\frac{1}{6}\).
### How These Probabilities Change with Different Number of Names
1. If the number of names in the hat decreases:
- The theoretical probability of any one name, including Josh, would increase. For example, if there were only 3 names, the theoretical probability would be \(\frac{1}{3}\).
- The experimental probability may still fluctuate based on how many times each name is randomly selected in actual trials.
2. If the number of names in the hat increases:
- The theoretical probability of any one name would decrease. For example, if there were 12 names, the theoretical probability would be \(\frac{1}{12}\).
- The experimental probability could change as well based on the outcomes during the trials—Josh might not be selected as much or could be selected more frequently, depending on chance over many trials.
In general, the experimental probability can vary greatly depending on the results of the specific trials, while the theoretical probability is fixed based on the number of names.