Events and Trials
An event is called random if, under a certain set of conditions $S$, it may or may not occur.
An event is the result of a trial.
Example
- A shooter fires at a target. The shot is a trial. Hitting or missing is an event.
Events are mutually exclusive if the occurrence of one of them precludes the occurrence of the others in the same trial.
Example
- A ball is drawn from a box. “A red ball is drawn” and “A blue ball is drawn” are mutually exclusive events;
- Getting heads and tails are mutually exclusive events.
Several events form a complete group if at least one of these events will occur as a result of a trial. A complete group of events is denoted by $\Omega$.
If the events forming a complete group are pairwise mutually exclusive, then only one of them can occur.
Example
- Heads and tails form a complete group of events
Two events are called complementary if they are the only two possible events forming a complete group. The event complementary to event $A$ is denoted $\bar{A}$.
Two events are called compatible if the occurrence of one does not preclude the occurrence of the other in the same trial.
Example
- A die is thrown. Event $A$ is rolling 4, and event $B$ is rolling an even number. Events $A$ and $B$ are compatible.
Classical Definition of Probability
Probability is a number characterizing the degree of likelihood of an event occurring.
Each possible outcome of a trial is an elementary outcome.
The probability of event $A$ is the ratio of the number of elementary outcomes favorable to event $A$ to the total number of outcomes, denoted $P(A)$.
$P(A) = \frac{m}{n}$, where $m$ is the number of favorable outcomes, $n$ is the total number of outcomes.
Example
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All possible elementary outcomes :* $\omega_1$ - white ball :* $\omega_2$ - white ball :* $\omega_3$ - black ball
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Events $A$ - a white ball is drawn $B$ - a black ball is drawn
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Probabilities $P(A) = \frac{2}{3}$ $P(B) = \frac{1}{3}$
Properties of Probability
- . Probability of a certain event $A$: $P(A) = 1$
- . Probability of an impossible event $A$: $P(A) = 0$
- . Probability of a random event $A$: $0 < P(A) < 1$
Thus, $0 \leqslant p \leqslant 1$
Statistical Definition of Probability
The classical definition of probability assumes a finite number of elementary outcomes. In practice, however, this number can be infinite. Also, it is often impossible to represent the result as a collection of elementary events.
Statistical definition of probability - the relative frequency $\frac{m}{n}$ is taken as the probability.
All properties of probability still hold: $0 \leqslant p = \frac{m}{n} \leqslant 1$
Principle of Practical Impossibility of Unlikely Events
If a random event has a very small probability, then in practice it can be considered that this event will not occur in a single trial.
Significance level - a sufficiently small probability at which an event can be considered impossible.
Sources
- Gmurman V.E., Probability Theory and Mathematical Statistics – 9th edition. Moscow: Vyssh. shk., 2003.