Experimental probability and theoretical probability are two different concepts, and while they can be close under certain conditions, they are not always guaranteed to align. Here's a breakdown of both:
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Theoretical Probability:
- This is calculated based on the possible outcomes in a perfect world. For example, the theoretical probability of rolling a 3 on a fair six-sided die is \( \frac{1}{6} \) because there’s one favorable outcome (rolling a 3) out of six possible outcomes.
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Experimental Probability:
- This is determined through actual experiments or trials. For instance, if you roll a die 100 times and roll a 3 15 times, the experimental probability of rolling a 3 would be \( \frac{15}{100} = 0.15 \).
Relationship between Experimental and Theoretical Probability:
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Convergence with Increasing Trials:
- The Law of Large Numbers states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. This means that if you conduct a very large number of trials, the experimental probability will likely converge to the theoretical probability.
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Variability in Small Samples:
- In smaller samples, experimental probability can vary significantly from the theoretical probability due to random chance. For example, if you only roll the die 10 times, you might not roll a 3 at all, resulting in an experimental probability of 0.
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Conditions:
- For the experimental probability to approach the theoretical probability, the trials must be conducted under controlled and fair conditions (e.g., a fair die, no bias in selection, etc.).
Conclusion:
In summary, while experimental probability can be close to theoretical probability with a large number of trials, it is not guaranteed to always be close, especially in smaller sample sizes. Random variability plays a significant role, and as the number of trials increases, the alignment between these two probabilities is more likely.