Probability refers to the greater or lesser possibility that an event will occur. His notion comes from the need to measure the certainty or doubt that a given event occurs or not. This establishes a relationship between the number of favorable events and the total number of possible events. For example, throwing a die, and the number one coming up (favorable case) is related to six possible cases (six heads); that is, the probability is 1/6.
What is probability
Table of Contents
It is the possibility of an event happening depending on the conditions given for it to happen (example: how likely is it to rain). It will be measured between 0 and 1 or expressed in percentages, said ranges can be observed in solved probability exercises. For this, the relationship between the favorable and possible events will be measured.
Favorable events are valid according to the experience of the individual; and the possible ones are those that can be given if they are valid or not in your experience. Probability and statistics are related to being the area where events are recorded. The etymology of the term comes from the Latin probabilitas or possitatis, related to "prove" or "verify" and tat which refers to "quality". The term relates to the quality of testing.
History of probability
It has always been in the mind of man, when they observed the possibility of some fact, for example, the diversity in the states of the climate based on the observation of natural phenomena to determine which possible climatic scenario could occur.
The Sumerians, Egyptians and Romans used the talus (heel bone) of some animals, to carve them in such a way that when thrown they could fall into four possible positions and what probability is there that they will fall into one or the other (like current dice). Tables were found where they allegedly made annotations of results.
Around 1660 a text came to light on the first foundations of chance written by the mathematician Gerolamo Cardano (1501-1576) and in the seventeenth century the mathematicians Pierre Fermat (1607-1665) and Blaise Pascal (1623-1662) tried to solve problems about games of chance.
Based on his contributions, the mathematician Christiaan Huygens (1629-1695) tried to explain the probabilities of winning a game and published on probability.
Later contributions such as Bernoulli's theorem, limit and error theorem and probability theory emerged, focusing on this Pierre-Simon Laplace (1749-1827) and Carl Frierich Gauss (1777-1855).
The naturalist Gregor Mendel (1822-1884) applied it to science, studying genetics and possible results in the combination of specific genes. Finally, the mathematician Andrei Kolmogorov (1903-1987) in the 20th century started the theory of probability that is known today (measure theory) and the probability statistics are used.
Probability measurement
Rule of addition
If we have an event A and an event B, its calculation would be expressed with the following formula:
taking into account that P (A) corresponds to the possibility of event A; P (B) would be the possibility of event B.
This expression means the possibility that anyone will occur.
This expression represents the possibility that both occur simultaneously.
Its exception is if the events are mutually exclusive (they cannot occur at the same time) because they do not have elements in common. An example would be the probability of rain, the two possibilities would be that it rained or not, but both conditions cannot exist at the same time.
With the formula:
Rule of multiplication
Both an event A and an event B occur simultaneously (joint probability), but it is subject to determining whether both events are independent or dependent. They will be dependent when the existence of one influences the existence of the other; and independent if they have no connection (the existence of one has nothing to do with the occurrence of the other). It is determined by:
Example: a coin is tossed twice, and the chance of the same heads coming up would be determined by:
so there is a 25% chance that the same face will appear both times.
Laplace rule
It is used to make estimates about the possibilities of an event that is not very frequent.
Determined by:
Example: Finding the percentage chance of drawing an Ace from a 52-piece deck of cards. In this case, the possible cases are 52 while the favorable cases 4:
Binomial distribution
It is a probability distribution where only two possible outcomes are obtained, known as success and failure. It must comply with: its possibility of success and failure must be constant, each result is independent, the two cannot occur simultaneously. Its formula is
where n is the number of attempts, x the successes, p probabilities of success and q probabilities of failure (1-p), also where
Example: if in a classroom 75% of the students studied for the final exam, then 5 of them meet. What is the probability that 3 of them have passed?
Types of probability
Classical probability
All possible cases have the same chance of happening. An example is a coin, in which the chances are the same that it comes up heads or tails.
Conditional probability
It is the probability that an event A occurs in knowledge that another B also happens and is expressed P (AB) or P (BA) as the case may be and it would be understood as “the probability of B given A”. There is not necessarily a relationship between the two or it may be that one is a consequence of the other, and they can even happen at the same time. Its formula is given by
Example: in a group of friends, 30% like the mountains and the beach, and 55% like the beach. What would be the probability that one who likes the beach likes the mountains? The events would be that one likes the mountains, another likes the beach, and one likes the mountains and the beach, so:
Frequency probability
The favorable cases are divided with the possible ones, when the latter tends to infinity. Its formula is
where s is the event, N the number of cases and P (s) the probability of the event.
Probability applications
Its application is useful in various areas and sciences. For example, probability and statistics are closely related, as well as with mathematics, physics, accounting, philosophy, among others, in which their theory helps to reach conclusions about possible eventualities and find methods to combine the events when multiple events are involved in a random experiment or test.
A palpable example is the prediction of weather conditions, games of chance, economic or geopolitical projections, probability of damage that an insurance company takes into account, among others.
