To find the experimental and theoretical probabilities of the name Ted being chosen, we'll follow the respective formulas for probability.
Part 1: Experimental Probability
The experimental probability (P) of an event is calculated using the formula:
\[ P(\text{event}) = \frac{\text{Number of times the event occurs}}{\text{Total number of trials}} \]
In this case, the event is choosing the name Ted.
Given:
- Number of times Ted is chosen = 27
- Total number of times names are chosen = 119
So the experimental probability of choosing Ted is:
\[ P(\text{Ted}) = \frac{27}{119} \]
Part 2: Theoretical Probability
The theoretical probability assumes that each name has an equal chance of being chosen. The theoretical probability of choosing the name Ted can be calculated as follows:
\[ P(\text{Ted}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
Given there are 6 different names, the theoretical probability of choosing Ted is:
\[ P(\text{Ted}) = \frac{1}{6} \]
Summary of Results
- Experimental probability of choosing the name Ted: \( \frac{27}{119} \)
- Theoretical probability of choosing the name Ted: \( \frac{1}{6} \)
Explanation of Changes with Different Number of Names in the Hat
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Experimental Probability: This is based on the actual outcomes of the trials. If the number of names were different, the experimental probability could change based on how many times each name is drawn in that new setting.
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Theoretical Probability: If the number of names in the hat were to change, the theoretical probability would also change. For example:
- If there were 4 names in the hat: \( P(\text{Ted}) = \frac{1}{4} \)
- If there were 10 names in the hat: \( P(\text{Ted}) = \frac{1}{10} \)
Thus, the theoretical probability is inversely proportional to the total number of names, while the experimental probability depends on actual results from trials conducted.