Many, faced with the concept of "probability theory", are frightened, thinking that this is something overwhelming, very complex. But it's really not all that tragic. Today we will consider the basic concept of probability theory, learn how to solve problems using specific examples.
Science
What does such a branch of mathematics as "probability theory" study? It notes patterns of random events and quantities. For the first time, scientists became interested in this issue back in the eighteenth century, when they studied gambling. The basic concept of probability theory is an event. It is any fact that is ascertained by experience or observation. But what is experience? Another basic concept of probability theory. It means that this composition 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 is simply a witness to these events, he does not influence what is happening in any way.
Events
We learned that the basic concept of probability theory is an event, but did not consider the classification. All of them are divided into the following categories:
- Reliable.
- Impossible.
- Random.
No matterwhat kind of events are observed or created in the course of experience, they are all subject to this classification. We offer to get acquainted with each of the species separately.
Certain event
This is 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 subject to this law. Probability theory includes such an important concept as a certain event. Here are some examples:
- We work and get remuneration in the form of wages.
- We passed the exams well, passed the competition, for this we receive a reward in the form of admission to an educational institution.
- We invested money in the bank, we will get it back if necessary.
Such events are reliable. If we have fulfilled all the necessary conditions, then we will definitely get the expected result.
Impossible events
Now we are considering elements of probability theory. We propose to move on to an explanation of the next type of event, namely, the impossible. First, let's specify the most important rule - the probability of an impossible event is zero.
You cannot deviate from this wording when solving problems. To clarify, here are examples of such events:
- Water froze at plus ten (that's impossible).
- The lack of electricity does not affect production in any way (just as impossible as in the previous example).
More examplesIt is not worth citing, since the ones described above very clearly reflect the essence of this category. The impossible event will never happen during the experience under any circumstances.
Random events
Studying the elements of probability theory, special attention should be paid to this particular type of event. That is what science is studying. As a result of experience, something may or may not happen. In addition, the test can be repeated an unlimited number of times. Vivid examples are:
- Tossing a coin is an experience, or a test, heading is an event.
- Blindly drawing a ball out of a bag is a test, a red ball is caught 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 type of all presented.
| title | definition | example |
| Reliable | Events that occur with a 100% guarantee under certain conditions. | Admission to an educational institution with a good entrance exam. |
| Impossible | Events that will never happen under any circumstances. | It is snowing at a temperature of plus thirty degrees Celsius. |
| Random | An event that may or may not occur during an experiment/test. | Hit or miss when throwing a basketball into the hoop. |
Laws
Probability theory is a science that studies the possibility of an event occurring. Like the others, it has some rules. There are the following laws of probability theory:
- Convergence of sequences of random variables.
- The law of large numbers.
When calculating the possibility of a complex, you 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 with the help of some theorems. Let's start with the first law.
Convergence of sequences of random variables
Note that there are several types of convergence:
- The sequence of random variables converges in probability.
- Almost impossible.
- RMS convergence.
- Convergence in distribution.
So, on the fly, it's very hard to get to the bottom of it. Here are some definitions to help you understand this topic. Let's start with the first look. 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 close to one.
Going to the next view, almost certainly. They say thatthe sequence converges almost surely to a random variable with n tending to infinity and P tending to a value close to one.
The next type is root-mean-square convergence. When using SC-convergence, the study of vector random processes is reduced to the study of their coordinate random processes.
The last type remains, let's take a brief look at it in order to proceed directly to solving problems. Distribution convergence has another name - “weak”, we will explain why below. Weak convergence is the convergence of distribution functions at all points of continuity of the limit 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.
Law of large numbers
Excellent helpers in proving this law will be theorems of probability theory, such as:
- Chebyshev's inequality.
- Chebyshev's theorem.
- Generalized Chebyshev's theorem.
- Markov's theorem.
If we consider all these theorems, then this question may drag on for several dozen sheets. Our main task is to apply the theory of probability in practice. We invite you to do this right now. But before that, let's 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 the impossible event. Let's remember: the probability of an impossible event is zero. We gave a very vivid and memorable example: it snowed at an air temperature of thirty degrees Celsius.
The second one sounds like this: a reliable event occurs with a probability equal to one. Now let's show how to write it using mathematical language: P(B)=1.
Third: A random event may or may not occur, but the possibility always ranges from zero to one. The closer the value is to one, the greater the chance; if the value approaches zero, the probability is very low. Let's write this in mathematical language: 0<Р(С)<1.
Let's consider the last, fourth axiom, which sounds like this: the probability of the sum of two events is equal to the sum of their probabilities. We write in mathematical language: P (A + B) u003d 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
First, 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 circulation, one of which has a prize of five hundred rubles, ten of one hundred rubles, fifty of twenty rubles, and one hundred of five. Problems in probability theory are based on finding the possibilitygood luck. Now together we will analyze the solution of the above presented task.
If we denote by the letter A a win of five hundred rubles, then the probability of getting A will be 0.001. How did we get it? You just need to divide the number of "lucky" tickets by their total number (in this case: 1/1000).
B is a win 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 winnings are equal to twenty rubles. Find the probability, it equals 0.05.
The rest of the tickets are of no interest to us, since their prize fund is less than the one specified in the condition. Let's 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 the previous steps. 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 can be more complex, for example, take the following task. Before you 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.
First, let's find the probability that the first card will be an ace, for this we divide four by thirty-six. They put it aside. We take out 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 drew first, we are interested inwas it an ace or not. It follows that event B depends on event A.
The next step is to find the probability of simultaneous implementation, that is, we multiply A and B. Their product is found as follows: the probability of one event is multiplied by the conditional probability of another, which we calculate, assuming that the first event occurred, that is, with the first card we drawn an ace.
To make everything clear, let's give a designation to such an element as the conditional probability of an event. It is calculated assuming that event A has occurred. Calculated as follows: P(B/A).
Continue solving our problem: P(AB)=P(A)P(B/A) or P (AB)=P(B)P(A/B). The probability is (4/36)((3/35)/(4/36). Calculate by rounding to hundredths. We have: 0, 11 (0, 09/0, 11)=0, 110, 82=0, 09. The probability that we draw two aces in a row is nine hundredths. The value is very small, it follows that the probability of the occurrence of the event is extremely small.
Forgotten number
We propose to analyze a few more options for tasks that are studied by probability theory. You have already seen examples of solving some of them in this article, let's 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 to the problem is the simplest if the rules, laws and axioms of probability theory are known.
Before watchingsolution, try to solve it yourself. We know that the last digit can be from zero to nine, that is, there are ten values in total. 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 immediately scored the right one, the probability of such an event is 1/10. The second option: the first call is a miss, and the second is on target. We calculate the probability of such an event: multiply 9/10 by 1/9, as a result we also get 1/10. The third option: the first and second calls turned out to be at the wrong address, only from the third the boy got where he wanted. We calculate the probability of such an event: we multiply 9/10 by 8/9 and by 1/8, we get 1/10 as a result. According to the condition of the problem, we are not interested in other options, so it remains for us to add up the results, as a result we have 3/10. Answer: The probability that the boy calls no more than three times is 0.3.
Cards with numbers
There are nine cards in front of you, on each of which a number from one to nine is written, the numbers are not repeated. They were placed in a box and mixed thoroughly. You need to calculate the probability that
- an even number will come up;
- two-digit.
Before proceeding to the solution, let's stipulate that m is the number of successful cases, and n is the total number of options. Find the probability that the number is even. It will not be difficult to calculate that there are four even numbers, this will be our m, there are nine options in total, that is, m=9. Then the probabilityequals 0, 44, or 4/9.
Consider the second case: the number of options is nine, and there can be no successful outcomes at all, that is, m equals zero. The probability that the drawn card will contain a two-digit number is also zero.
