Question

Challenge  Six different names were put into a hat. A name is chosen 112 times and the name Karen is chosen 25 times. What is the experimental probability of the name Karen being​ chosen? What is the theoretical probability of the name Karen being​ chosen? Use pencil and paper. Explain how each probability would change if the number of names in the hat were different.
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Part 1
The experimental probability of choosing the name Karen is
  
enter your response here.
​(Type an integer or a simplified​ fraction.)

Answer 1

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

- 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.

- 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!