The difference between experimental probability and theoretical probability lies in how they are calculated and the type of data they use:

1. Theoretical Probability:

   • Definition: This type of probability is based on the assumption of ideal conditions and is calculated using mathematical principles.
   • Formula: The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
   • Example: When flipping a fair coin, the theoretical probability of landing heads is: $P(\text{Heads}) = \frac{1}{2} \quad \text{or} \quad 0.5$
   • Use: It is used when the outcomes are well-known, and there is no variability due to real-world factors.

2. Experimental Probability:

   • Definition: This type of probability is based on actual experiments or observations. It is calculated by performing an experiment or conducting trials and recording the outcomes.
   • Formula: The probability of an event is determined by dividing the number of times the event occurs by the total number of trials. $P(E) = \frac{\text{Number of times event E occurs}}{\text{Total number of trials}}$
   • Example: If you flip a coin 100 times and it lands heads 55 times, the experimental probability of getting heads is: $P(\text{Heads}) = \frac{55}{100} = 0.55$
   • Use: It is used when you cannot predict or rely on perfect conditions, and you must experiment to observe actual outcomes.

Key Differences:

• Calculation: Theoretical probability is calculated from known information or assumptions, while experimental probability is based on actual data from trials.
• Accuracy: Theoretical probability assumes ideal conditions, while experimental probability can be affected by real-world variability, biases, or errors.
• Dependence on Trials: Experimental probability can improve as more trials are conducted (the Law of Large Numbers), while theoretical probability does not change with the number of trials.

Would you like more details or further examples on this topic?

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Here are 5 related questions you might be interested in:

1. How does the Law of Large Numbers affect experimental probability?
2. Can experimental and theoretical probabilities ever be the same?
3. What are some examples of situations where theoretical probability might not apply well?
4. How do you calculate probability in complex experiments with multiple outcomes?
5. What are the limitations of using experimental probability in real-world scenarios?

Tip: For a more accurate estimate of experimental probability, always conduct as many trials as possible to minimize variability.