Frequency and Probability
Big Idea
Probability can be understood in two fundamental ways: theoretical probability (what we expect to happen based on logic) and experimental probability (what actually happens when we perform experiments). Frequency is the bridge that connects real-world data to probability.
Key Definitions
Frequency: The number of times a particular outcome occurs in an experiment or observation.
Relative Frequency: The ratio of the frequency of a particular outcome to the total number of trials.
$$ \text{Relative Frequency} = \frac{\text{Frequency of the outcome}}{\text{Total number of trials}} $$
Experimental (Empirical) Probability: The probability of an event estimated from the relative frequency of the event in an experiment.
$$ P(E) = \frac{f(E)}{n} $$
where $f(E)$ is the frequency of event $E$ and $n$ is the total number of trials.
Theoretical Probability: The probability calculated using logical reasoning about equally likely outcomes.
$$ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} $$
Properties of Probability
- Range: For any event $E$, $0 \leq P(E) \leq 1$
- Certain Event: $P(\text{certain event}) = 1$
- Impossible Event: $P(\text{impossible event}) = 0$
- Complement Rule: $P(\text{not } E) = 1 - P(E)$
- Sum of All Probabilities: The sum of the probabilities of all possible outcomes equals 1: $$ \sum_{i=1}^{k} P(E_i) = 1 $$