Probability is the number of chances that a particular event will occur.
This figure is the result of a numerical calculation. It is normally expressed using whole numbers (it is common to use figures from 1 to 10) or percentages (which have the number 100 as a common denominator).
For example, if we roll a six-figure die, we can say that there is a 1 in 6 chance that the number 5 will come up. But we could also say that there is a 16.6% chance that we will roll a 5 (the result of dividing 100 between 6).
What Is Probability?
The science that measures and studies probabilities is statistics.
The events for which we can determine probabilities are of any condition. Therefore, probability can be used in practically all fields of life. But some, by their very nature, are more susceptible to becoming the object of statistics. Among them are the game, weather phenomena, the results of an election, etc.
In addition, there are events whose probability of occurring can be quantified more easily.
The formulas to calculate the probability are several and depend on the event and the type of probability that we need to know. These are the main ones:
Main types of probability
Classical or mathematical probability
It is the result of dividing the number of favorable cases (number of times an event occurs) by the total number of possible cases. This is a theoretical calculation, based on mathematical logic and not on experience.
Therefore, it adapts very well to events or situations whose number of possibilities is easily calculable, such as the roll of a die, the choice of a card within a deck, etc.
It is known as a priori probability or logical probability.
The basic formula to calculate it is:
Understanding by P (Probability), by S, the number of favorable cases and by N, the total number of possible cases.
With this formula we can calculate the probability that the number we have chosen will appear in French roulette (which consists of 37 numbers), using this same formula.
It is also known as frequentist and is based on the relative frequency with which an event actually occurs (empirically). That is, the proportion of times it happens within a series of identical or similar repetitions.
Therefore, an event is considered “probable” if it tends to occur with some regularity as the experiment is repeated many times.
It is a type of probability that can only be measured if the necessary conditions are met to do so. This means that in the experiment you can control all the variables and observe how the results develop in a large number of repetitions.
We can make use of frequency probability when it comes to knowing the probability of rain in a certain place at a certain time of the year. For which all the rainfall that occurred year after year on the same date in the same place must have been measured.
Subjective probability is that which is calculated through formulas or experiments, but also includes elements based on the opinion or personal judgment of the person performing the experiment. This occurs because, sometimes, there are non-quantifiable elements that can interfere with the probability of an event happening.
Unlike frequentist or mathematical interpretations, which are based on empirical observations or formal calculations, subjective probability reflects the uncertainty perceived by a person based on their own knowledge and experience.
We could use subjective probability if we wanted to measure the chances of a basketball player making a basket from a point x on the playing field.
To do this we could use a frequency probability experiment, but consider that as the player performs the experiment he gains in skill. Therefore, we could “infer” or “assume” that the more times we perform the experiment, the more the player’s probability of scoring will increase.
It measures the probability of two events happening simultaneously. That is, they happen at the same time.
In the context of two events (X, Y), the joint probability would be measured by the following formula.
Formula and example:
With this function we could measure, for example, the chances that when tossing a coin and throwing a 6 die the result would be heads and 6 at the same time.
Which equates to an 8.33% chance of the two events happening simultaneously.
It is the calculation of the possibilities that exist for a given subset of elements to appear within a larger set.
Marginal probability allows us to calculate the probability of an event in a particular variable without taking into account the possible combinations of values of other variables. This can be especially relevant when the objective is to know the statistical probability of a specific variable, without the need to analyze the probability of other similar variables.
Example and formula to calculate it
The general formula for the marginal probability of an event X is calculated by the sum of the times that event has occurred divided by the total number of data.
For example, if we write down whether people have been seasick during a ship’s journey depending on the weather that day for thirty days, we would have:
- Days with sun-sick people: 3
- Days with people dizzy with rain: 9
- Days with non-sunsick people: 11
- Days with people not dizzy with rain: 7
Therefore, four out of every ten days there will be someone dizzy.
Conditional or conditional probability measures the possibilities that exist for an event to occur, once another has already happened.
It is based on the idea that we are restricting the sample space to event X. In the new sample space we consider, we evaluate the probability of Y, relative to the possibilities within that reduced sample space.
Formula and examples
The mathematical formula is:
For example, if 30% of your family members like to play tennis and soccer, and 50% like to play soccer. What are the chances that a member of your family who likes to play soccer also likes tennis?
Therefore, the probability that a member of your family also likes tennis is 60%.
It is based on Bayes’ rule and measures the probability of a hypothesis based on a given new hypothesis. It is a form of probabilistic reasoning that allows us to adjust the probability of a fact based on the appearance of a new one.
The key formula in Bayesian probability is Bayes’ rule:
- P(X/Y) is the probability that X happens after Y has happened.
- P(Y/X) is the probability that Y happens after X has happened.
- P(X) is the probability that X happens.
- P(Y) is the probability that Y happens.
For example, in a class there is a 25% chance that a student will like Don Quixote, and a 30% chance that they will like La Celestina. The fact is also known that if a student likes Celestina, the possibility of her liking Don Quixote is 50%. Bayesian probability helps us know the percentage of chances that a student who likes Don Quixote will like La Celestina.
- P(X/Y) is the probability that you like Don Quixote if you like La Celestina.
- P(Y/X) is the probability that you like La Celestina if you like Don Quixote.
- P(X) is the probability that you like Don Quixote.
- P(Y) is the probability that you like La Celestina.
Therefore, the probability that a student likes Don Quixote having liked La Celestina is 41%.
It allows us to know the probability that an element is found at a given point in a given space.
It is measured by dividing the length of the segment that we seek to know by the total length of the space.
Formula and example OF PROBABILITY
Given a 10 cm segment. of length, A and C being their ends and B a point on that line. The distance between A and B is 3 centimeters. If we need to calculate the probability that a point is on part of the segment AB, the probability will be the length of that part of the segment divided by the total length of the segment.
The probability that the point is in the part of segment AB is 30%.
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