Algebra of Random Events
Definition 1. Events are called mutually exclusive if the occurrence of one excludes the occurrence of others in the same trial. Events are called compatible if the occurrence of one does not exclude the possibility of others occurring.
Example 1. Among homogeneous parts in a box, there are standard and non-standard parts. One part is randomly taken from the box.
Events:
$A$ - the taken part is standard,
$B$ - the taken part is non-standard,
are mutually exclusive because only one part is taken, which cannot be both standard and non-standard.
Example 2. Two shooters shoot at a target.
Events:
$A_1$ - the first shooter hits the target,
$A_2$ - the second shooter hits the target,
are compatible random events.
Definition 2. Random events $A_1, A_2, \ldots, A_n$ form a complete group of events if at least one of them must occur as a result of the trial.
Example 3. A six-sided die is rolled. Let the events be denoted as follows:
$A_1$ - side 1 appears;
$A_2$ - side 2 appears;
$A_3$ - side 3 appears;
$A_4$ - side 4 appears;
$A_5$ - side 5 appears;
$A_6$ - side 6 appears.
Events $A_1, A_2, \ldots, A_6$ form a complete group.
Note that in Example 2, events $A_1$ and $A_2$ do not form a complete group. However, if we denote by $A_3$ the event that neither shooter hits the target, and by $A_4$ the event that both shooters hit the target, then events $A_1, A_2, A_3$, and $A_4$ form a complete group.
Definition 3. Events are called equally likely in some experiment if there is no reason to believe that any one of them is more likely than the others.
Example 4. The events of obtaining 1, 2, 3, 4, 5, or 6 points when rolling a six-sided die are equally likely provided the center of gravity is not displaced.
Definition 4. Two mutually exclusive events that form a complete group are called complementary. An event complementary to event $A$ is denoted by $\bar{A}$ (Fig. 1.1).
Example 5. If $A$ denotes the event that 8 points are scored when shooting at a target, then $\bar{A}$ denotes the event that any other number of points is scored.
Now consider the important concept of the sample space.
Let an experiment involving elements of randomness be performed, i.e., each trial can have different outcomes. For instance, tossing a coin can result in either heads or tails. Rolling a die can result in six possible outcomes. In the experiment "shooting at a target," outcomes such as hitting the target, the number of points scored, or the coordinates of the hit can be considered.
Definition 5. Elementary outcomes are those that cannot be broken down into simpler ones. The set of all possible elementary outcomes is called the sample space.
The sample space may contain a finite, countable, or uncountable set of elements. Elementary outcomes can be points in an $n$-dimensional space, a segment of a line, points on the surface $S$, or volume $V$ of a three-dimensional space, a function of one or many variables. In most cases considered, elementary outcomes are assumed to be equally likely.
Let $A$ and $B$ be random events.
Definition 6. The union $A + B$ (or $A \cup B$) of random events $A$ and $B$ is a random event that occurs if either event $A$ or event $B$ or both occur. Thus, the union of two events is the event that occurs if at least one of the events $A$ or $B$ occurs. If $A$ and $B$ are mutually exclusive, then $A + B$ denotes the occurrence of either event $A$ or event $B$. Similarly, the union of more than two random events is defined.
Definition 7. The intersection $A \cdot B$ (or $A \cap B$) of two random events $A$ and $B$ is a random event that occurs if both events $A$ and $B$ occur simultaneously. If $A$ and $B$ are mutually exclusive, then $A \cdot B = \emptyset$.
Definition 8. The intersection $A_1 \cdot A_2 \cdot \ldots \cdot A_n$ of random events $A_1, A_2, \ldots, A_n$ is a random event that occurs if all these events occur simultaneously.
These relationships between events can be represented graphically (see Fig. 1.1).
Example 6. A shooter fires twice at a target. Describe the sample space. Write the event that:
a) the shooter hits exactly once (event $C$);
b) the shooter hits the target at least once (event $D$);
c) the shooter misses the target (event $F$).
Solution:
Let $A$ be the event of hitting the target on the first shot, and $B$ be the event of hitting the target on the second shot. The sample space consists of four events: $$\{AB, A\bar{B}, \bar{B}A, \bar{A}\bar{B}\}$$
a) The shooter hits exactly once, which can occur if the shooter hits on the first shot and misses on the second, or misses on the first shot and hits on the second. Thus,
$$C = A * \bar{B} + \bar{B} * A$$
b) The shooter hits the target at least once, which means hitting on the first shot ( $A$ ), or hitting on the second shot ( $B$ ), or hitting on both shots ($A \cdot B$). Thus,
$$D = A + B$$
c) The shooter misses the target, which means missing on both shots. Thus,
$$F = \bar{A} * \bar{B}$$
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