Probability it is a branch of mathematics in which the chances of experiments occurring are calculated. It is through a probability, for example, that we can know from the chance of getting heads or tails on a coin flip to the chance of error in polls.
To understand this branch, it is extremely important to know its most basic definitions, such as the formula for the probability calculation in equiprobable sample spaces, probability of the union of two events, probability of the complementary event etc.
random experiment
is any experience whose result is not known. For example: when flipping a coin and looking at the top side, it is impossible to know which side of the coin will facing up, except in the case where the coin is biased (modified to have a more often).
Suppose a grocery bag contains green and red apples. Taking an apple out of the bag without looking is also a experimentrandom.
Sample point
One Scoresample is any possible result in a experimentrandom. For example: on the roll of a die, the result (the number that appears on the top face) can be 1, 2, 3, 4, 5 or 6. So each of these numbers is a sampling point for this experiment.
Sample space
O sample space it's the set formed by all sample points on one random experiment, that is, for all its possible results. In this way, the result of a random experiment, even if it is not predictable, can always be found within the sample space referring to it.
Like the spacessample are sets of possible outcomes, we use set representations for these spaces. For example: The sample space referring to the experiment “rolling a die” is the set Ω, such that:
Ω = {1, 2, 3, 4, 5, 6}
That set can also be represented by the venn diagram or, depending on the experiment, by some formation law.
O numberinelements of the sample spaces is represented by n (Ω). In the case of the previous example, n (Ω) = 6. Remember that the elements of a sample space are pointssample, that is, possible results of a random experiment.
Event
Events are subsets of a spacesample. One event it can contain from zero to all possible results of a random experiment, that is, the event can be an empty set or the sample space itself. In the first case, it is called impossible event. In the second, it's called right event.
not yet experimentrandom of rolling a die, note the following events:
A = Get an even number:
A = {2, 4, 6} and n (A) = 3
B = Leave a prime number:
B = {2, 3, 5} and n (B) = 3
C = Exit a number greater than or equal to 5:
C = {5, 6} and n (C) = 2
D = Leave a natural number:
D = {1, 2, 3, 4, 5, 6} and n (D) = 6
Equiprobable spaces
A sample space is called equiprobable when all pointssample within it have the same chance of occurring. This is the case of unaddicted dice rolls or coins, choosing numbered balls of identical size and weight, etc.
An example of spacesample that can be considered not equiprobable is formed by the following experiment: choose between having ice cream or going for a walk.
Probability Calculation
At odds are calculated by dividing the number of favorable outcomes by the number of possible outcomes, ie:
P = huh)
n (Ω)
In this case, E is an event that one wants to know the probability, and Ω is the spacesample that contains it.
For example, on the roll of a die, what is the probability of the number one coming out?
In this example, exiting number one is event E. Thus, n (E) = 1. The sample space of this experiment contains six elements: 1, 2, 3, 4, 5 and 6. Therefore, n (Ω) = 6. Thus:
P = huh)
n (Ω)
P = 1
6
P = 0.1666…
P = 16.6%
Another example: what is the probability to get an even number when rolling a die?
The possible even numbers on a die are 2, 4 and 6. Hence, n (E) = 3.
P = huh)
n (Ω)
P = 3
6
P = 0.5
P = 50%
Note that the odds will always result in a number within the range 0 ≤ x ≤ 1. This is because E is a subset of Ω. In this way, E can contain from zero to at most the same number of elements as Ω.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/probabilidade.htm