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Frequencies, probabilities and calculation rules

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Here you will learn how to determine probabilities in random experiments, what Laplace experiments are and how relative frequencies are related to probabilities.

Random experiments, probabilities and probability spaces

Random experiments are used for the mathematical description of processes, the outcome of which cannot be predicted with certainty. For example, when you roll the dice, you cannot know beforehand whether you will roll a 1, 2, 3, 4, 5 or 6. In these cases, probabilities are intended to measure and indicate the degree of certainty that events will occur. Numbers are used for this and assigned to the events: The impossible event has a probability of 0, it never occurs. The certain event has the probability 1, it always occurs. Events with different probabilities sometimes, but not always, occur.
A probability is specified with a number: it is at least 0 and at most 1 and is denoted by P.
If you randomly draw one of 5 different cards (numbered from 1 to 5), card 4 is drawn with a probability of 0.2.
In some situations, probabilities are also given in percentages or ratios. You can convert as usual, for example: The weather forecast on television reports: "The probability of rain tomorrow is 70."
Determining result sets Before the probabilities of events can be calculated, it must be clarified which result set is being considered. Probabilities are only assigned to events that belong to the result set, that is, that are a subset of the result set. To do this, you use the letter P (probability). Often there are several ways to specify a result set. Depending on the event the probability of which you want to determine, you will need to choose it cleverly. In this explanation only experiments in which the result set is finite are considered.
Which amount do you have to choose as the result set if you want to specify the probability with which the number 2 will fall when you roll the dice?
Probabilities of events A result set Ω together with an assignment of probabilities P for every possible event is also called a probability space. The events of a probability space are often named with Latin capital letters, mostly E.
Max throws a coin in the air, which will land either on the heads or on the tails side with a probability of 0.5. The result set is Ω = {heads; Number}. Let E be the event “lands on the number side”, so E = {number}. What are the probabilities of the following events?
Determine probabilities

General rules for probabilities

If the probabilities of all results in a probability space are known, then the probabilities of any associated events can also be calculated. Conversely, probabilities of results can sometimes be inferred from probabilities of events. These relationships are summarized in the so-called sum rule: If there is an event E = {; , ...; } from the pairwise different results,, ...,, then the probability of E corresponds to the sum of the probabilities of the results: P (E) = P ({}) + P ({}) +… + P ({}) .
The sum of the probabilities of an event and its counter-event is always 1.
The sum of the probabilities of two different results a and b always corresponds to the probability of the event “a or b”: P ({a}) + P ({b}) = P ({a; b}).
If you draw a ticket at the fair with a probability of 0.1 you get a winning ticket and with a probability of 0.3 you get a consolation ticket, you will draw a winning or a consolation ticket with a probability of 0.4:    
Probability of the counter-event For every event E of a probability space there is a counter-event. This contains exactly the results that are not contained in E. If, for example, Ω = {a; b; c; d; e} and E = {a; b; c}, then = {d; e}. The following applies to the sum of the probabilities of the event E and its counter-event
During the elf, each pupil in class 8b draws a ticket with the name of the classmate he is going to give, and Mika draws one of the tickets first. His teacher has calculated that there is a 0.48 probability that he will draw a ticket with a maiden name; what is the probability that he will draw a ticket with a boy's name?
Determine probability
Mika draws a ticket with the name of a boy with a probability of 0.52.
Application of the sum rule If the probability of an event is known, which for example consists of two outcomes, and the probability of one of the outcomes is also known, then the probability of the other outcome can also be deduced from this.
A lottery seller advertises: “Only 10 of the lots are rivets.” In addition to the rivets, the lottery pot only contains lots for consolation prizes and lots for main prizes. What is the probability that you will win a consolation prize if 15 of the lots are jackpot prizes?
Determine probability
With a chance of you will win a consolation award.

Laplace probabilities and the resulting rules

A random experiment of a probability space with result set Ω is called a Laplace experiment if every result a ∈ Ω is equally likely. You can recognize Laplace experiments mostly from existing symmetries, for example the shape of a thrown object (dice, coin) or the arrangement of winning fields like on a roulette wheel.
In a Laplace experiment the following applies to the probability P of an event: P =
Roll the dice with result set Ω = {1; 2; 3; 4; 5; 6} is a Laplace experiment because when rolling a “fair” cube, due to the symmetrical shape and the evenly distributed mass of the cube, each of the six sides remains on top with the same probability. Each number is therefore thrown with probability. Roll the dice with result set Ω = {no 6; 6} is not a Laplace experiment because a “6” is less likely than “no 6”, ie one of the five other results.
Determining Laplace probabilities The condition that every result must be equally probable, together with the generally applicable sum rule, follows a rule with which Laplace probabilities can often be easily calculated. For example, in a Laplace experiment, if Ω = {1; 2; 4; 6; 8} the result set, every result must occur with probability. For the probability of the event E = {4; 6} then follows: P (E) = P ({4}) + P ({6}) = + = = In order to calculate the probability of an event, in Laplace experiments only the ratio of the number of contained results to the number of all possible results has to be determined .
Formula: In a Laplace experiment with the result set Ω, the probability of an event E: P (E) = =
Give the probability for the event E = {rot; yellow} in a Laplace experiment with result set Ω = {green; blue; red; yellow; black; White; pink} on.
Laplace probability
If you want to determine a probability of an event in an experiment that is described in a word problem, you first have to determine what the result set is or how you can choose it and whether it is a Laplace experiment or not.
In roulette, a ball is rolled into a wheel of fortune with fields numbered from 0 to 36 and evenly spaced. What is the probability of hitting the 10, 20 or 30 with the ball? First select which of these sets you can define as the result set Ω in order to determine the desired probability in a Laplace experiment.
Since there are 37 different possibilities for the result and 3 of them are favorable for the event described, the probability we are looking for is P ({10; 20; 30}) =.

Relative frequencies

The relative frequency of a property in observations indicates the ratio of the number of observations with this property to the total number of all observations. It describes a proportion that is at least 0 and at most 1. As a proportion or ratio, a relative frequency is represented as a decimal number, but often also in percent or as a fraction.
Relative frequency of A =
During soccer training, Marc converted 18 of 25 penalties. Fabian has converted 16 of 20 penalties. Marc: - Number of all penalties: 25 - Absolute frequency of hits: 18 - Relative frequency of hits: = = 72 Fabian: - Number of all penalties: 20 - Absolute frequency of hits: 16 - Relative frequency of hits: = = 80
In this statistic, 530 members of an association are divided into three age categories: - 104 members are 28 years of age or younger. - 201 members are older than 28 years and at most 45 years old. - 225 members are over 45 years old. Enter the relative frequencies in this table.
Determine relative frequencies
Estimate absolute frequency
The age of 106 other club members is not known. Assume that the age structure of all 636 members matches that of the 530 members from the statistics and estimate with the help of the relative frequencies how many of the 636 club members are over 45 years old. According to the calculated relative frequency, 270 club members are over 45 years old.
Estimating probabilities If no exact probability can be determined for an event of a random experiment, a probability can be estimated by performing this experiment as often as possible. The observed relative frequency is then used for this: If an event occurs in, for example, 960 of 2000 identical experiments, it is assumed that its probability in this experiment is approximately = 48. Conversely, it is assumed that the relative frequencies of events come closer and closer to the actual probabilities as the number of attempts increases. These assumptions are justified by the observation that in very many random experiments the relative frequencies fluctuate less and less with an increasing number of attempts. This is also called the empirical law of large numbers. It cannot be proven generally and is therefore not a mathematical proposition. It is more a fact of experience and only indicates that random occurrences also follow certain regularities.
A car manufacturer uses a so-called crash test dummy as the driver of the new model to simulate a rear-end collision 2000 times and after each attempt checks whether a person would have been injured as the driver. These are the counted absolute frequencies: According to these statistics, what is the probability that a driver of this car will be injured in a comparable accident?
Estimate probability
Since a driver would have been injured in 684 of the 2000 tests, the probability to be estimated using this statistic is P (injury) = = 34.2.

Expected values

If a number is assigned to each possible result in a random experiment, it can be calculated which number is assigned on average when the experiment is carried out frequently. This average result is called the expected value. You determine the expected value of a random experiment with regard to an assignment Z of numbers by multiplying the assigned number by the probability of the result for each of the possible results, ... and then calculating the sum of these products.
Expected value = Z () P () + Z () P () + ... + Z () P ()
A wheel of fortune with 4 fields of the same size is spun. One is yellow, one blue and two are green. Turning blue gives you € 3 profit, turning green brings you € 1 profit, turning yellow gives you nothing (€ 0). Z assigns the corresponding profit in euros to each result: Z (blue) = 3Z (green) = 1Z (yellow) = 0 Since each of the four fields is hit with probability, the probabilities of the three results are: P (blue) = P (green) = P (yellow) = The following applies to the expected value of this experiment: