Question
is experimental probability always close to theoretical probability?
Answers
Answered by
GPT-4o mini
Experimental probability is not always close to theoretical probability, especially in smaller sample sizes. The key differences between the two are:
1. **Definition**:
- **Theoretical Probability** is based on the known outcomes of an experiment, calculated using the formula:
\[
P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\]
- **Experimental Probability** is based on the actual results obtained from conducting an experiment. It is calculated using the formula:
\[
P(E) = \frac{\text{Number of times event E occurs}}{\text{Total number of trials}}
\]
2. **Sample Size**:
- With a small number of trials, the experimental probability can vary significantly from the theoretical probability due to random chance. As the number of trials increases, the experimental probability tends to converge to the theoretical probability, a principle known as the Law of Large Numbers.
3. **Variability**:
- Experimental results can be influenced by numerous factors such as biases in the experiment, external conditions, and random variation, which can also lead to discrepancies between the two probabilities.
In conclusion, while experimental probability often approaches theoretical probability as the number of trials increases, it is not guaranteed to be close, especially in smaller samples.
1. **Definition**:
- **Theoretical Probability** is based on the known outcomes of an experiment, calculated using the formula:
\[
P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\]
- **Experimental Probability** is based on the actual results obtained from conducting an experiment. It is calculated using the formula:
\[
P(E) = \frac{\text{Number of times event E occurs}}{\text{Total number of trials}}
\]
2. **Sample Size**:
- With a small number of trials, the experimental probability can vary significantly from the theoretical probability due to random chance. As the number of trials increases, the experimental probability tends to converge to the theoretical probability, a principle known as the Law of Large Numbers.
3. **Variability**:
- Experimental results can be influenced by numerous factors such as biases in the experiment, external conditions, and random variation, which can also lead to discrepancies between the two probabilities.
In conclusion, while experimental probability often approaches theoretical probability as the number of trials increases, it is not guaranteed to be close, especially in smaller samples.