What is the Difference Between Theoretical and Experimental Probability?
Answer: Theoretical probability relies on mathematical analysis, using the ratio of favorable outcomes to possible outcomes, whereas experimental probability is derived from observed outcomes in real-world trials.
What is Theoretical Probability?
Theoretical probability is based on mathematical analysis and relies on the assumption of equally likely outcomes in a sample space. It is calculated using the following formula:
P(E) = Number of Favorable Outcomes/Total Number of Possible Outcome
Where:
- P(E) is the probability of event E.
- The number of favorable outcomes is determined through mathematical reasoning.
What is Experimental Probability?
Experimental probability, on the other hand, is derived from actual observations or experiments. It involves conducting trials or experiments and recording the outcomes to determine the probability. The formula for experimental probability is:
P(E) = Number of Favorable Outcomes in Experiment/Total Number of Trials or Experiments
Where:
- P(E) is the experimental probability of event E.
- The number of favorable outcomes is observed through experimentation.
Difference Between Theoretical and Experimental Probability
The following table gives the tabular difference between Theoretical and Experimental Probability:
| Feature | Theoretical Probability | Experimental Probability |
|---|---|---|
| Definition | Based on mathematical analysis and reasoning. | Based on observations and empirical data. |
| Calculation | Calculated using mathematical formulas. | Determined by conducting experiments or trials and observing outcomes. |
| Prediction | Provides an idealized prediction of probability. | Represents a real-world approximation of probability. |
| Formula | 𝑃(𝐸)=Number of favorable outcomesTotal number of possible outcomesP(E)=Total number of possible outcomesNumber of favorable outcomes | 𝑃(𝐸)=Number of times event E occurredTotal number of trialsP(E)=Total number of trialsNumber of times event E occurred |
| Example | Flipping a fair coin: Theoretical probability of getting heads is 1221. | Rolling a fair six-sided die: Experimental probability of getting a 5 after 100 rolls is 0.17. |
| Application | Commonly used in theoretical mathematics and probability theory. | Commonly used in experimental sciences and real-world situations where outcomes can be observed. |
| Assumptions | Assumes all outcomes are equally likely. | May involve assumptions about randomness and the conditions of the experiments. |
| Accuracy | Perfectly accurate under ideal conditions. | May be subject to errors due to limitations in sample size or biases in the experiment. |