[Basics of Probability Theory] Sample Space and Event | Sample Space and Event

Foreword:

Personal study notes, since they are foreign textbooks, the translated mathematical terms may be slightly different from those in domestic textbooks.

content

Foreword:

0x00 Definition

0x01 Recovery extraction and non-recovery extraction

0x02 sum event and post event

0x03 Extra and bad events

0x04 Mutually exclusive events and pairwise mutually exclusive events

0x05 partition (partition)

0x00 Definition

probability

The degree of likelihood of an event occurring under certain conditions is expressed in numerical form and is called probability.

statistics

Various data ranging from social phenomena to natural phenomena are analyzed and expressed in numerical form according to a certain system.

random phenomenon

In social and natural phenomena, the result is not a predetermined phenomenon, that is, the phenomenon that the structure is affected by some uncertainty, which is called a probabilistic phenomenon.

random experiment

Experiments, observations, or investigations in which the results are expressed as probabilistic phenomena are called probabilistic experiments. (Example: flip a coin, flip a dice, draw a card)

sample space

The set of all possible outcomes that can be obtained from a probability experiment, called probability is the sample space of the experiment, denoted by the symbol $S$ or $\Omega$ .

sample point

Each element of the specimen space is called a sample point, denoted by $w_1, w_2, w_3...$ etc.

event

A partial set (subset) of the specimen space, a set of specific sample points that meet certain conditions, is called an event, $A,B,C$ and is represented by capital letters, etc.

total event

Specimen space $\Omega$ is full event.

empty event

Record as $\O$ .

elementary event

An event consisting of one sample point is called a root event.

refer to:

① The sample space in the coin toss probability experiment is:

$\Omega = \left \{ H, T \right \}$ (The head of the coin is H, the tail is T)

② The sample space for a single throw of the dice is:

example:

① Find the sample space in the probability experiment of tossing a coin three times $\Omega$ .

$\Omega = \left \{ \left ( i,j \right ) | 1 \leq i,j\leq 6 \right \} =$

② Request event A with at least one positive (H) appearance.

$A = \left \{ \left ( i,j \right ) \in \Omega | i=j \right \} =$

③ Request the event B that the back (T) appears more times than the front (H).

$B = \left \{\left ( i,j \right )\in \Omega | i=2j \right \}=$

0x01 Recovery extraction and non-recovery extraction

When repeating the same experiment multiple times in a probability experiment:

Recovery extraction (replacement)

The way to return the extracted content and extract the next one, we call it recovery extraction. (replacement)

(replace sampling)

Extraction without replacement (without replacement)

The way of extracting the next one without returning the extract is called a non-recovery extract.

(no replacement sampling)

0x02 sum event and post event

$\Omega$ For two events A and B of a partial set of specimen space :

and event (union of event)

A combined event of A and B: an event where event A or event B occurs

intersection of events

Intersection of A and B: An event where event A and event B occur at the same time is called an intersection event

refer to:

0x03 Extra and bad events

$\Omega$ For two events A and B of a partial set of specimen space :

complementary event

Events A did not happen:

difference of event

An event where event A occurs but event B does not:

0x04 Mutually exclusive events and pairwise mutually exclusive events

Mutually exclusive events

When the two events A and B in the partial set of the specimen space $\Omega$ do not occur at the same time, then the two events AB are called mutually exclusive events.

pairwisely mutually exclusive events

$\Omega$ For a partial set (subset) of $n$ events in the specimen space $A_1, A_2, ... ,A_n$ ,

, $A_1, A_2, ... ,A_n$ is called a paired mutually exclusive event.

(any two are mutually exclusive)

0x05 partition (partition)

The following two conditions for $\Omega$ a partial set (subset) of $n$ events in the specimen space : $A_1, A_2, ... ,A_n$

① $A_1, A_2, ... ,A_n$ is a pair of mutually exclusive events. but

② If satisfied , we call it the division of specimen space $\Omega$ (complete event group).

That is, several events in the experiment, they are mutually exclusive and at least one event occurs.

Example: Toss a coin three times. If the event that defines the number of heads appears $i$ , please prove that the event is the division $A_i(i=0,1,2,3)$ of the specimen space . $\Omega$