Definition and Properties of Probability and Frequency
To compare random events by their degree of possibility, each event must be associated with a certain number, which should be larger for more likely events. This number $P$ is called the probability of the event. There are several definitions of probability. Let's review them.
Definition. The probability of an event is a numerical measure of the degree of objective possibility of that event. This definition captures the philosophical essence of probability but does not specify the rule for finding the probability of any event.
Classical Definition. The probability of an event $A$ is the ratio of the number of elementary outcomes favorable to event $A$ to the total number of equally likely and mutually exclusive elementary outcomes that form a complete group.
The probability of event $A$ is denoted as $P(A)$. By definition,
$$P(A) = \frac{m}{n}$$
where $m$ is the number of elementary outcomes favorable to event $A$, and $n$ is the total number of equally likely and mutually exclusive elementary outcomes that form a complete group.
Example 1. An urn contains 6 balls of equal size: 2 red, 3 blue, and 1 white. Find the probability of drawing a red ball if one ball is drawn at random from the urn.
$$P(red ball) = \frac{2}{6} = \frac{1}{3}$$
Frequency Definition. The probability of an event $A$ is the limit of the relative frequency of this event in a large number of trials, provided this limit exists:
$$P(A) = \lim\limits_{n\rightarrow \infty}\frac{m}{n}$$
where $m$ is the number of times event $A$ occurs in $n$ trials.
Properties of Probability:
The probability of a certain event is 1:
$$P(\Omega) = 1$$
where $\Omega$ denotes the sample space.
The probability of an impossible event is 0:
$$P(\emptyset) = 0$$
For any event $A$, the probability $P(A)$ is a number between 0 and 1:
$$0 ≤ P(A) ≤ 1$$
If two events $A$ and $B$ are mutually exclusive, the probability of their union is the sum of their probabilities:
$$P(A \cap B) = P(A) + P(B)$$
Example 2. Consider a standard deck of 52 playing cards. What is the probability of drawing either a king or a queen?
$$P(king) = \frac {4}{52} = \frac {1}{13}$$
$$P(queen) = \frac {4}{52} = \frac {1}{13}$$
Since drawing a king and drawing a queen are mutually exclusive events,
$$P(king or queen) = P(king) + $P(queen) = \frac {1}{13} + \frac {1}{13} = \frac {2}{13} $$
Definition (Geometric). The probability of event $A$ is equal to the ratio of the measure of $g$ to the measure of $G:$
$$ P(A) = \frac{m(g)}{m(G)}$$
Remark 3. If the region $G$ is an interval, surface, or spatial body, and $g$ is a part of $G$, then the measure of $G$ and $g$ will be the length, area, or volume of the corresponding region. If $G$ and $g$ are time intervals, then their measure will be time.
Definition. The relative frequency or frequency of event $A$ is called the ratio of the number of independent trials in which event $A$ occurred to the number of actually conducted independent trials.
The relative frequency of event $A$ is usually denoted as $\nu(A)$. Hence,
$$W(a) = \frac{m}{n}$$
where $m$ is the number of independent trials in which event $A$ occurred, and $n$ is the number of all independent trials.
Example 3. The technical control department found 8 non-standard items among 100 products. What is the relative frequency of non-standard items?
Solution. Let $A$ denote the event of finding a non-standard item. Then, by the definition of the frequency of an event, we obtain:
$$W(A) = \frac{8}{100} = 0,08$$
Remark 4. Note that the probability $P(A)$ of event $A$ is calculated before the trial, whereas the frequency $\nu(A)$ is calculated after the trial.
The frequency of the occurrence of an event has the property of stability: with a large number of trials, the frequency changes very little, oscillating around a certain constant (number) – the probability of the occurrence of this event, i.e.,
$$P(A) = \lim\limits_{n\rightarrow \infty}W(A)$$
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