Frequency and Probability
Preparation for the Grade 9 final state exam (IGA) — Probability Theory and Statistics
The main idea
Probability is a predicted (theoretical) quantity, while frequency is a statement of fact (the result of an observation). When the number of trials is large, the relative frequency approaches the probability of the event.
Classical (theoretical) probability
The probability of an event $A$ is the ratio of the number $m$ of outcomes favourable to this event to the total number $n$ of all equally likely mutually exclusive outcomes:
$$P(A) = \frac{m}{n}$$
- $m$ — the number of favourable outcomes (those in which event $A$ occurs)
- $n$ — the total number of all equally likely outcomes
Important: The numerator cannot be greater than the denominator, so a probability always satisfies the inequality: $$0 \leq P(A) \leq 1$$
Special cases
| Event | Probability |
|---|---|
| Certain (always happens) | $P = 1$ |
| Impossible (never happens) | $P = 0$ |
| Random (may or may not happen) | $0 < P < 1$ |
Complementary (opposite) events:
If $A$ and $\bar{A}$ are complementary events, then: $$P(\bar{A}) = 1 - P(A)$$ Their probabilities add up to 1.
If $A$ and $\bar{A}$ are complementary events, then: $$P(\bar{A}) = 1 - P(A)$$ Their probabilities add up to 1.