Experimental vs. Theoretical Probability
Big Idea
Probability measures how likely an event is to occur. But there are two fundamentally different ways to determine this likelihood: by reasoning mathematically (theoretical) or by conducting experiments (experimental). Understanding both — and the relationship between them — is essential for statistics, data science, and real-world decision making.
Theoretical Probability
Definition
Theoretical probability is the probability of an event determined by mathematical reasoning, assuming all outcomes are equally likely. It is calculated before any experiment is performed.
$$ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{|E|}{|S|} $$
Key Characteristics
- Based on logic and symmetry, not data
- Requires knowledge of the sample space $S$
- Assumes equally likely outcomes (classical probability)
- Is an exact value — it doesn't change between calculations
- Example: $P(\text{heads on a fair coin}) = \frac{1}{2} = 0.5$
Experimental Probability
Definition
Experimental probability (also called empirical probability or relative frequency) is the probability of an event determined by actually performing an experiment and recording the results.
$$ P(E) = \frac{\text{Number of times event } E \text{ occurs}}{\text{Total number of trials}} = \frac{f(E)}{n} $$
Key Characteristics
- Based on observed data from actual trials
- Does not require knowledge of the sample space
- Varies between different sets of trials
- Becomes more reliable as the number of trials increases
- Example: If you flip a coin 100 times and get 47 heads, $P(\text{heads}) = \frac{47}{100} = 0.47$