Many, when confronted with the concept of "theory of probability," are frightened, thinking that it is something overwhelming, very complex. But everything is actually not so tragic. Today we will consider the basic concept of probability theory, and learn how to solve problems using concrete examples.
The science
What is studying such a branch of mathematics as “probability theory”? She notes the patterns of random events and quantities. For the first time, scientists became interested in this issue in the eighteenth century, when they studied gambling. The basic concept of probability theory is an event. This is any fact that is ascertained by experience or observation. But what is experience? Another basic concept of probability theory. It means that this set of circumstances was not created by chance, but for a specific purpose. As for observation, here the researcher himself does not participate in the experiment, but simply is a witness to these events, he does not affect what is happening.
Events
We learned that the basic concept of probability theory is an event, but we did not consider the classification. All of them are divided into the following categories:
- Reliable.
- Impossible.
- Random.
No matter what events they are observed or created during the course of the experiment, they are all subject to this classification. We offer you to get acquainted with each of the species separately.
Authentic event
This is such a circumstance, before which the necessary set of measures has been taken. In order to better understand the essence, it is better to give a few examples. Physics, chemistry, economics, and higher mathematics are subordinate to this law. Probability theory includes such an important concept as a reliable event. Here are some examples:
- We work and receive remuneration in the form of wages.
- We passed the exams well, passed the competition, for this we get a reward in the form of admission to the educational institution.
- We have invested money in the bank, if necessary we will get it back.
Such events are reliable. If we have met all the necessary conditions, then we will definitely get the expected result.
Impossible events
Now we are considering the elements of probability theory. We suggest moving on to explaining the next type of event, namely, the impossible. To begin with, we will stipulate the most important rule - the probability of an impossible event is zero.
One cannot deviate from this formulation when solving problems. For clarification, we give examples of such events:
- Water froze at a temperature of plus ten (this is impossible).
- Lack of electricity does not affect production in any way (as impossible as in the previous example).
More examples should not be given, as the ones described above very clearly reflect the essence of this category. An impossible event will never happen during an experience under any circumstances.
Random events
Studying the elements of probability theory, special attention should be paid to this particular type of event. It is them that this science is studying. As a result of experience, something can happen or not. In addition, the test can be carried out an unlimited number of times. Bright examples are:
- A coin toss is an experience, or a test; an eagle is an event.
- Pulling the ball out of the bag blindly is a test, a red ball is caught - this is an event and so on.
There can be an unlimited number of such examples, but, in general, the essence should be clear. To summarize and systematize the knowledge gained about events, a table is given. Probability theory studies only the last of all presented.
title | definition | example |
Authentic | Events occurring with a one hundred percent guarantee subject to certain conditions. | Admission to an educational institution with a good entrance exam. |
The impossible | Events that will never happen under any circumstances. | It snows at air temperature plus thirty degrees Celsius. |
Random | An event that may or may not occur during an experiment / test. | Hit or miss when throwing a basketball in the ring. |
The laws
Probability theory is a science that studies the possibility of an event occurring. Like others, she has some rules. The following laws of probability theory exist:
- Convergence of sequences of random variables.
- The law of large numbers.
When calculating the possibility of a complex one, one can use a complex of simple events to achieve the result in an easier and faster way. Note that the laws of probability theory are easily proved using some theorems. We suggest first to get acquainted with the first law.
Convergence of sequences of random variables
Note that there are several types of convergence:
- The sequence of random variables is similar in probability.
- Almost impossible.
- RMS convergence.
- Convergence in distribution.
So, on the fly, it’s very difficult to understand the essence. We give definitions that will help to understand this topic. For starters, the first view. A sequence is called convergent in probability if the following condition is met: n tends to infinity, the number to which the sequence tends is greater than zero and is close to unity.
We move on to the next view, almost certainly . It is said that the sequence converges almost certainly to a random variable with n tending to infinity, and P tending to a value close to unity.
The next type is RMS convergence . When using SC convergence, the study of vector random processes reduces to the study of their coordinate random processes.
There is the last type, let's analyze it briefly, so that we can go directly to solving problems. Convergence in distribution has another name - “weak”, then we explain why. Weak convergence is the convergence of distribution functions at all points of continuity of the limiting distribution function.
Be sure to fulfill the promise: weak convergence differs from all of the above in that the random variable is not defined on the probability space. This is possible because the condition is formed exclusively using distribution functions.
The law of large numbers
Probability theory theorems, such as:
- Chebyshev inequality.
- Chebyshev theorem.
- A generalized Chebyshev theorem.
- Markov theorem.
If we consider all these theorems, then this question can drag on to several tens of sheets. Our main task is to apply probability theory in practice. We offer you to do this right now. But before that, we consider the axioms of probability theory, they will be the main assistants in solving problems.
Axioms
We already met the first one when we talked about an impossible event. Let's remember: the probability of an impossible event is zero. An example we cited was very vivid and memorable: snow fell at an air temperature of thirty degrees Celsius.
The second is as follows: a reliable event occurs with a probability equal to one. Now we show how to write it using the mathematical language: P (B) = 1.
Third: A random event may or may not occur, but the possibility always varies from zero to one. The closer the value is to unity, the greater the chances; if the value approaches zero, the probability is very small. We write this in mathematical language: 0 <P (C) <1.
Consider the last, fourth axiom, which sounds like this: the probability of the sum of two events equals the sum of their probabilities. We write in mathematical language: P (A + B) = P (A) + P (B).
The axioms of probability theory are the simplest rules that are easy to remember. Let's try to solve some problems, based on the knowledge already gained.
Lottery ticket
To begin, consider the simplest example - the lottery. Imagine that you bought one lottery ticket for good luck. What is the probability that you will win at least twenty rubles? In total, a thousand tickets participate in the draw, one of which has a prize of five hundred rubles, ten for one hundred rubles, fifty for twenty rubles, and one hundred for five. Probability problems are based on finding the opportunity for luck. Now we will analyze the solution above the task presented.
If we denote a letter of five hundred rubles with the letter A, then the probability of A falling out will be 0.001. How did we get this? It is just necessary to divide the number of “happy” tickets by their total number (in this case: 1/1000).
B is a gain of one hundred rubles, the probability will be 0.01. Now we acted on the same principle as in the previous action (10/1000)
C - the gain is equal to twenty rubles. We find the probability, it is 0.05.
The rest of the tickets are not of interest to us, since their prize pool is less than the one specified in the condition. We apply the fourth axiom: The probability of winning at least twenty rubles is P (A) + P (B) + P (C). The letter P denotes the probability of the occurrence of this event, we have already found them in previous actions. It remains only to add the necessary data, in the answer we get 0,061. This number will be the answer to the question of the task.
Card deck
Probability theory problems are also more complicated, for example, take the following task. Here is a deck of thirty-six cards. Your task is to draw two cards in a row without mixing the pile, the first and second cards must be aces, the suit does not matter.
To begin with, we find the probability that the first card will be an ace, for this we divide four by thirty-six. Put it aside. We get the second card, it will be an ace with a probability of three thirty-fifths. The probability of the second event depends on which card we pulled out first, we wonder if it was an ace or not. It follows that event B depends on event A.
By the next action we find the probability of simultaneous implementation, that is, we multiply A and B. Their product is found as follows: we multiply the probability of one event by the conditional probability of the other, which we calculate, assuming that the first event happened, that is, the first card we pulled out an ace.
In order to make everything clear, we give a designation to such an element as the conditional probability of an event. It is calculated, assuming that event A has occurred. It is calculated as follows: P (B / A).
We continue the solution to our problem: P (A * B) = P (A) * P (B / A) or P (A * B) = P (B) * P (A / B). The probability is (4/36) * ((3/35) / (4/36). We calculate it by rounding to the nearest hundredth. We have: 0.11 * (0.09 / 0.11) = 0.11 * 0, 82 = 0.09 The probability that we will draw two aces in a row is nine hundredths, the value is very small, it follows that the probability of the origin of the event is extremely small.
Forgotten number
We suggest analyzing several more options for tasks that probability theory studies. Examples of solutions to some of them you have already seen in this article, we will try to solve the following problem: the boy forgot the last digit of his friend’s phone number, but since the call was very important, he began to dial everything in turn. We need to calculate the probability that he will call no more than three times. The solution is the simplest if the rules, laws and axioms of probability theory are known.
Before you watch the solution, try to solve it yourself. We know that the last digit can be from zero to nine, that is, only ten values. The probability of getting the right one is 1/10.
Next, we need to consider options for the origin of the event, suppose that the boy guessed right and scored the right one, the probability of such an event is 1/10. The second option: the first call miss, and the second to the target. We calculate the probability of such an event: 9/10 we multiply by 1/9, as a result we also get 1/10. The third option: the first and second calls were not at the address, only from the third the boy got to where he wanted. We calculate the probability of such an event: 9/10 multiply by 8/9 and 1/8, we get in the end 1/10. Other options for the conditions of the problem do not interest us; therefore, it remains for us to add up the results obtained, as a result we have 3/10. Answer: the probability that the boy will call no more than three times is 0.3.
Numbers Cards
Before you are nine cards, on each of which is written a number from one to nine, the numbers are not repeated. They were put in a box and mixed thoroughly. You need to calculate the probability that
- an even number will appear;
- two-digit.
Before proceeding to the solution, we stipulate that m is the number of successful cases, and n is the total number of options. Find the probability that the number will be even. It will not be difficult to calculate that there are four even numbers, this will be our m, in total there are nine possible options, that is, m = 9. Then the probability is 0.44 or 4/9.
We consider the second case: the number of options is nine, but there can be no successful outcomes at all, that is, m is zero. The likelihood that an extended card will contain a two-digit number is also equal to zero.