What are the principles of probability

Probability calculation: formulas, examples and explanations

The calculation of probability - often also called stochastics - is one of the worst chapters in mathematics for most students. In the following you will find an overview of the topics that we would like to cover here. This is followed by a brief introduction to the most important topics.

Probability theory or probability theory is an extensive chapter in math. So I have divided the topic into different topics. First we look at important basic terms and then we turn to topics such as the binomial coefficient, the urn model and much more. In this area, it is also important to get to know terms such as number of eyes, amount of events and much more. In many cases, exercises with solutions can be found at the end of each chapter.

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The event tree of probability theory

Many people wish to be able to predict events. Just a small example: "heads or tails?" it is often said when a coin is tossed. Of course, you don't know which of the two sides the coin will land on. Only one probability can be given. There are two sides: heads or tails. So the probability for coat of arms is 1/2 and for coin also 1/2. And that brings us to the event tree. We draw the example of the probability calculation with the coin from above in an event tree. There are two possibilities (coat of arms, number) that can occur with a throw, consequently there are two paths. The probability is 1/2 for coat of arms and 1/2 for number, these values ​​are written on the paths. But see for yourself:

All possibilities that exist can be combined into a result set "M". In our case this would be: M = {coat of arms, number}. Now of course we are interested in what actually happens in a real experiment. Take a look at the following table, which is explained below.

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Probability calculation: Laplace's rule

Let's come to another topic in the area of ​​probability theory: Let's first clarify the term random experiment: A random experiment is a process in which at least two results are possible and in which the result cannot be predicted before the process has ended. Example: A die is thrown. Which side he will land on cannot be said before the die is thrown from the hand. The random experiment therefore belongs to the field of probability theory.

A Laplace experiment is a random experiment in which all the possibilities of the outcome of the experiment are same probability exhibit. One often speaks here of "equally likely".

Laplace experiment: examples

How can you tell whether it is a Laplace experiment or not? The question is often not that easy to answer and in many cases requires prior knowledge in the relevant area. Here are a few examples:

  • A normal one cube has six sides. If nothing has been manipulated on the dice, the probability of rolling the number 1 is just as great as the probability of rolling the number 6. It is therefore a Laplace experiment.
  • A coin has two sides: heads and tails. If the coin has not been manipulated, the probability of throwing a "number" is just as great as the probability of throwing a "coat of arms". It is therefore a Laplace experiment.
  • In a horse race, 10 riders and horses compete against each other. Since the skills of the participants differ from each other, the chance of a victory is different for each participant. So we don't have a Laplace experiment.

One should try to approach such tasks with some common sense. If one has no reason to consider the occurrence of any one of the results of a random experiment as more likely than that of the other results, one can initially assume a Laplace experiment.

Binomial coefficient of the calculus of probability

The binomial coefficient of the calculus of probability is a mathematical function with which one of the basic problems of combinatorics can be solved. The binomial coefficient indicates the number of different ways in which k objects can be selected from a set of n different objects. The experiment is carried out without replacing and regardless of the sequence.

Calculate binomial coefficient

Let us now come to the notation for the binomial coefficient and its calculation. To do this, you need to know how to calculate the faculty (what is faculty?). In the following you will find the notation and its calculation. There are explanations afterwards.

Explanation: On the left side you will find the short notation for the binomial coefficient, spoken "n over k". On the right side you can see the fraction as it is calculated. The following examples should clarify this.

Example 1:

Random experiments

Let's start with the definition of the term random experiment: A random experiment is a process in which at least two results are possible and in which the result cannot be foreseen before the process has ended. Example: A die is thrown. Which side he will land on cannot be said before the die is thrown from the hand

One step random experiment

A one-step random experiment in probability calculation is a random experiment that is only carried out once. Examples:

  • A die is thrown once
  • A coin is tossed once

In most cases it is necessary to carry out an experiment several times. For example, the number 4 could be rolled when rolling a dice. But after a try one could believe that the number 4 is always thrown on a dice. For this reason, one-step random experiments are in most cases not meaningful. Therefore, in the following, we will take a closer look at the multi-stage random experiment or the multi-stage random experiment.

Multi-stage random experiment

A multi-stage random experiment is when a random process is carried out several times in a row. Example: A cube becomes multiple thrown in a row. If a multi-stage random experiment consists of k partial experiments, one speaks of a k-stage random experiment. The outcome of a random experiment is called the result. The result set contains all possible results of a random experiment.

Urn model of the calculus of probability

An urn model in probability calculus is a "box" in which there are balls. Balls are now pulled out of the box - without looking inside - and their number is noted. There are basically two different attempts:

  1. Put the urn model back with you: A ball is drawn from the urn. The number is written down and then the ball is thrown back into the urn. The number of balls in the urn therefore remains the same.
  2. Urn model without putting back: A ball is drawn from the urn. The number is written down and then the ball is thrown away. With each drawing, the number of balls in the urn is reduced.

My articles on probability theory:

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