Probability is the study of experiments that, even carried out under very similar conditions, present results that are not possible to predict. For example, the heads or tails experiment, even if performed repeatedly, cannot be predicted, because each time the coin is flipped, the result it might be different.
Probability associates numbers with chances of determined result happen, so that the higher this number, the greater the chance of this result occurring. There is a "small number", which represents the impossibility of result, and a larger number, which represents the certainty of a given result. When rolling a single die, for example, it is impossible for the number 7 to occur and there is certainty that a number less than 7 or greater than 0 will occur.
The most important definitions for the study of odds are the following:
Sample point
given one random experiment, any result only one of this experiment is called sample point.
When rolling two dice at the same time, the possible results they are:
1 and 1, 1 and 2, 1 and 3 … 6 and 5, 6 and 6
When tossing a coin, the sampling points are heads or tails.
Sample space
Sample space it's the set who owns all sample points on one random event. Therefore, the sample space referring to the experiment “flipping a coin” is formed by heads and tails.
O sample space it is also commonly called the universe. Also, as it is a set, any set notation can represent you.
In this way, the sample space, its subsets and the operations that involve it inherit the properties and operations of the numerical sets. Thus, we can say that the possible results of tossing two coins are:
S = {(x, y) natural | x < 7 and y < 7}
In this case, S represents the set of ordered pairs formed by the results of the two dice. The number of elements in a sample space is represented as follows: Given the sample space Ω, the number of elements of Ω is n (Ω).
Event
One event is any subset of a sample space. Thus, the events are formed by sampling points. An example of event is this: on the roll of two dice, only odd numbers should appear.
The subset that represents this event has the following sample points:
(1, 1)
(3, 3)
(5, 5)
they are the possible results of rolling two dice with odd results simultaneously.
The number of elements of an event is represented as follows: Given event A, the number of elements of A is n (A).
Also, an event is called a simple event when it has only one element, that is, when the event is equal to only one sample point. In other words, single event represents a single result. One right event is equal to the sample space, so the probability that a certain event will occur is the highest of all: 100% chance. On the other hand, when the event is equal to the empty set, that is, it does not have any sample point, he is called impossible event.
Probability
THE probability is a number that represents the chance an event has of happening. The calculation of this number is done as follows: let A be one event any inside the sample space Ω, the probability P(A) of this event happening is given by:
P(A) = at)
n (Ω)
Note, first of all, that the number of elements in the sample space will always be greater than or equal to the number of elements in the event. In this way, the smallest value this division can result is 0, which represents the chance that there is an impossible event. The highest value that can be reached is 1, when the event is the same as sample space. In this case, the result of the division is 1. In this way, the probability of an event A within the sample space Ω occur is between the range:
0 ≤ P(A) ≤ 1
There are two observations to make:
If it is necessary to express the probability on one event happen by means of a percentage, just multiply the result of the above division by 100.
There is the possibility to calculate the probability of an event not happening. To do so, just perform:
PAN-1) = 1 - P(A)
conditional probability
Given the sample space Ω and events A and B in Ω, assume that event A has already occurred. The probability that event B will occur is called conditional probability of B over A and is denoted as follows:
P(B|A)
That probability gets its name because the condition for B to occur is the occurrence of A. The expression used to calculate this probability is as follows:
P(B|A) = P(B)∩THE)
PAN)
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-probabilidade.htm
