To calculate both the experimental and theoretical probabilities for the name Karen being chosen, we'll break down the calculations step by step.
Part 1: Experimental Probability
The experimental probability (P) is calculated using the formula:
\[ P(\text{Karen}) = \frac{\text{Number of times Karen is chosen}}{\text{Total number of times names are chosen}} \]
Given the data:
- Number of times Karen is chosen = 25
- Total number of times names are chosen = 112
Substituting the values into the formula:
\[ P(\text{Karen}) = \frac{25}{112} \]
The experimental probability of choosing the name Karen is:
\[ \frac{25}{112} \]
This fraction is already in its simplest form, so we leave it as it is.
Part 2: Theoretical Probability
The theoretical probability (P) is based on the assumption that all names have an equal chance of being selected. Since there are 6 different names, the theoretical probability of choosing Karen is:
\[ P(\text{Karen}) = \frac{1}{\text{Total number of names}} = \frac{1}{6} \]
Conclusion: Changes with Different Numbers of Names
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If the total number of names in the hat were to increase, the theoretical probability of selecting Karen would decrease because the denominator (total number of names) would become larger, which makes the probability smaller.
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If the total number of names were to decrease (but still more than 1), the theoretical probability of selecting Karen would increase because the denominator would be smaller, resulting in a higher probability.
To summarize:
- Experimental Probability (from data): \(\frac{25}{112}\)
- Theoretical Probability (equal chance): \(\frac{1}{6}\)
Feel free to ask if you need further clarifications!