Probability theory. Basic terms and concepts

First and very important. What does this science study? Many thought for sure came thoughts like "the probability of rain is great", "the probability of winning the lottery is small," "the eagle and tails fall with a probability of 50 to 50," etc. But then the question immediately arises: where does science come from? Please, right now pick up the coin and tell me, which side will it fall after the throw? ... It's not at all like a theory - it's more like some guessing ....

And indeed, the philistine understanding of probability looks more like a prediction, often with a fair amount of mysticism and superstition. The probability theory studies probabilistic regularities of mass homogeneous random events . That is, she has no purpose to guess anything, for example, the result of throwing the same coin in a single experiment. However, if the same coin under the same conditions is tossed hundreds and thousands of times, then a clear pattern will be observed, described by quite strict laws.

Another example. Around each of us, air molecules fly. Some of them are high, some medium, and some - low speed. It makes no sense to guess the speed of individual molecules; but their mass accounting finds the widest application in theoretical and applied physical research. Note that the planes "know how" to fly, gas and steam boilers do not usually explode, and the kettles do not jump in the kitchen when boiling. For many and many seemingly ordinary facts and events, there are serious probabilistic and statistical calculations.

Or an easier example. If you get a lottery ticket, it's unlikely that you will win anything and it's quite incredible that you will lose a large jackpot. But the organizer of the lottery even with an occasional drawing of the draw (retrieval of numbered balls, etc. or if the participants themselves guess the numbers) is guaranteed and with a high degree of accuracy knows how many tickets will win / lose and, of course, remains in profit. Lotteries are often called deception, but the paradox is that this guarantee is strictly justified by theory ; Ravno, as well as the everyday phrase "still do not win anything." I think now everyone has understood the correct way of earning on lotteries =) However, we will return to the "secrets" of winning roulette and various lotteries.

Yes, by the way, think about one more pressing task: many of us take dozens of exams for life, and practically always the following situation occurs: a part of the questions the student knows (or the spurs are prepared), and some questions - do not know (or swim as a master of sports) . Day "X" comes: morning, a corridor with 10-15 classmates and a door, behind which there is a complete set of tickets on the table. In what case is it more likely to take the exam - if you go "in the front row", "in the middle" or if you go to the audience among the latter? ... We study the theory of probability!

First we deal with the basic terms, which I will outline in bold italics below . I draw your attention to the fact that these are TERMS, not "just words"!

Developments. Types of events
One of the basic concepts of the Terran has already been voiced above - this is an event . Events are reliable , impossible and random .

A certain event is called a certain event, which as a result of the test (the implementation of certain actions, a certain set of conditions) will necessarily occur . For example, in an earthly gravity, a tossed coin will certainly fall down.

Impossible is called an event, which obviously will not happen as a result of the test. An example of an impossible event: in the conditions of gravity, the tossed coin will fly upward.

And, finally, the event is called random , if as a result of the test it can, as it happens, never happen , and there must be a fundamental randomness criterion : a random event is a consequence of random factors whose impact is impossible or extremely difficult to predict. Example: as a result of throwing a coin, an "eagle" will drop out. In the case considered, random factors are the shape and physical characteristics of the coin, the force / direction of the throw, air resistance, etc.

The accentuated randomness criterion is very important - for example, a card sharper can very skillfully imitate randomness and give a win to the victim, but there is no question of any random factors affecting the final result.

Any test result is called an outcome , which, in fact, represents the appearance of a certain event. In particular, when flipping a coin, there are probably 2 outcomes (random events): an eagle will fall out, a tail falls out. Naturally, it is understood that this test is conducted in such conditions that the coin can not stand on the edge or, say, hang in weightlessness.

Events (any) represent capital letters or by the same letters with subscripts, for example . The exception is a letter that is reserved for other needs.

We write the following random events:

• as a result of a coin toss, an "eagle" will drop out;
• As a result of a dice roll (cube), 5 points will be dropped;
• A card of the club's color will be removed from the deck (by default, the deck is considered complete) .

Yes, events are written directly in practical tasks, and in appropriate cases it is convenient to use "speaking" subscripts (although you can do without them).

It should be emphasized for the third time that random events necessarily satisfy the above randomness criterion. In this sense, the third example is again indicative: if from the deck initially all the clubs of the club suit are deleted, then the event becomes impossible . Conversely, if the tester knows that, for example, the lady of clubs is at the bottom, then he can, if he wishes, make the event reliable =) Thus, in this example it is assumed that the cards are well mixed and their shirts are indistinguishablei.e. The deck is not crooked. And, here under the "crap" is meant not even "skilful hands" that eliminate the chance of your winnings, but the visible defects of the cards. For example, the shirt of the same ladies of clubs can be dirty, torn, sealed with scotch tape ... pancake, some benefit for a beginner chikatilo turned =)

Thus, when drawing an important lot, there is always a sense to see if the sides of the coin are the same ;-)

Another important characteristic of events is their equal opportunity . Two or more events are called equally possible , if none of them is more possible than others. For example:

the fall of an eagle or tails during a coin toss;
loss of 1, 2, 3, 4, 5 or 6 points in the roll of the dice;
Extraction of a card of a club, peak, diamonds or hearts from the deck.

It is assumed that the coin and the cube are homogeneous and have a geometrically correct shape, and the deck is well mixed and "perfect" from the point of view of the indistinguishability of card shirts.

Can the same events be not equally possible ? They can! For example, if a coin or a cube has a center of gravity displaced , then quite definite faces will fall out more often. As they say, another loophole for scammers. Events - the extraction of clubs, peaks, hearts or diamonds are also equally possible. However, the magician can equally easily violate this, which, shuffling the deck (even "ideal"), deftly peeps and hides in the sleeve, for example, an ace of clubs. Here it becomes less possible that the opponent will be given a club, and, most importantly, it is less likely that an ace will be dealt.

Nevertheless, in the three cases considered, if the probability is lost, the randomness of events is still preserved.

Joint and incompatible events. Opposite events.
Full Event Group
Events are called incompatible , if in the same test the occurrence of one of the events excludes the appearance of other events. The simplest example of inconsistent events is a pair of opposite events. The event opposite to this is usually denoted by the same Latin letter with a dot at the top. For example:

• as a result of a roll of the coin, an eagle will fall out;
• as a result of a coin drop, a tailshot falls out.

It is made clear that in a single trial the appearance of an eagle excludes the appearance of a tails (and vice versa), therefore these events are called incompatible.

Opposite events are easily formulated from considerations of elementary logic:

• as a result of the roll of the dice, 5 points will fall;
• as a result of the roll of the dice, the number of points different from five will fall out.

Either five, or not five - the third is not given, i.e. events are inconsistent and opposite .

Similarly - either a club or a card of another suit:

• from the deck will be drawn a map of the club suit;
• A peak, a heart or a tambourine will be extracted from the deck.

A number of incompatible events form a complete group of events , if as a result of a single test one of these events will necessarily appear . Obviously, any pair of opposite events (in particular, the examples above) form a complete group. However, different events can appear in different tasks with the same object, for example, for a game cube, the following set of features is typical:

• as a result of the roll of the dice, 1 point will drop;
• ... 2 points;
• ... 3 points;
• ... 4 points;
• ... 5 points;
• ... 6 points.

Events are incompatible (since the appearance of a face excludes the simultaneous appearance of others) and form a complete group (since as a result of the test one of these six events will inevitably appear) .

Another important concept that we will soon need is the elementality of the outcome (event). If very simple, then the elementary event "can not be decomposed into other events". For example, events are elementary, but the event is not so, as it implies a loss of 1, 2, 3, 4 or 6 points (includes 5 elementary outcomes).

In the example with event cards (extracting clubs, spades, hearts or tambourines, respectively) are incompatible and form a complete group, but they are non-elementary. If we assume that there are 36 cards in the deck, then each of the listed events includes 9 elementary outcomes. Similarly, events (extraction of the six, seven, ..., king, ace) are inconsistent, form a complete group and are not elementary (each includes 4 outcomes).

Thus, the elementary outcome here is only the extraction of a particular map, and, of course, 36 incompatible elementary outcomes also form a complete group of events.

Joint events are less significant from the point of view of solving practical problems, but we will not bypass them. Events are called joint , if in a single trial the appearance of one of them does not exclude the appearance of the other. For example:

• From the deck of cards will be extracted clubs;
• Seven of the cards will be removed.

If one is quite concise, one does not exclude the other.

The concept of compatibility covers a larger number of events:

• tomorrow at 12.00 it will rain;
• Tomorrow at 12.00 there will be a thunder-storm;
• tomorrow at 12.00 there will be sun.

The situation, of course, is quite rare, but the joint appearance of all three events in principle is not ruled out. It should be noted that the listed events are joint and in pairs, i.e. can only be a downpour with a thunderstorm or a mushroom rain, or pogromyhayet nearby in the background of a clear sky.

Algebra of events
Be brave, there will be matan =)

Please, remember the IMPORTANT RULE , without which mastering the terrain is simply unrealistic:

The operation of addition of events means a logical connection OR ,
and the operation of multiplication of events is a logical connective AND .

1. The sum of the two events and called the event which consists in the fact that there will come or event , or an event or two events simultaneously. In that case, if the events are mutually exclusive , the latter disappears, that is, can occur or an event , or an event .

The rule extends to a greater number of terms, for example, the event consists in that at least one of the events occurs , and if the events are inconsistent , then one and only one event from this sum: either an event , or an event , or an event , or an event , or event .

Examples of weight:

Events (when throwing dice 5 will not drop points) is that the rolled or 1, or 2, or 3, or 4, or 6 points.

The event (it will not be more than two points) is that there will be 1 or 2 points .

Event (to be an even number of points) is that the rolled or 2 or 4 or 6 points.

The event is that a card of the red suit (a worm or a tambourine) will be drawn from the deck , and the event is that the "picture" (jack or queen or king or ace) will be removed .

A little more amusing is the case with joint events:

The event is that from the deck will be extracted a club or seven or seven clubs. According to the above definition, at least something - or any clubs or any seven or their "intersection" - the seven clubs. It is easy to calculate that this event corresponds to 12 elementary outcomes (9 treasure maps + 3 remaining sevens).

The event is that tomorrow at 12.00 there will come at least one of the summable joint events , namely:

• or there will be only rain / only a thunderstorm / only the sun;
• or only a couple of events will come (rain + rain / rain + sun / thunderstorm + sun);
• or all three events will appear simultaneously.

That is, the event includes 7 possible outcomes.

The second pillar of the algebra of events:

1. The product of two events and called the event , which is co-occurrence of these events, in other words, multiplication means that come under some circumstances , and events , and event . A similar statement is true for a large number of events, for example, the work implies that under certain conditions happen and the event , and the event , and the event , ... and event .

Consider a test in which two coins and the following events are tossed :

• on the 1st coin, an eagle will fall;
• on the 1st coin a tails will drop out;
• on the 2nd coin, an eagle will fall;
• on the 2 nd coin will fall tails.

Then:

• the event consists in the fact that on both coins (on the 1st and 2nd) the eagle will fall out;
• the event consists in the fact that on both coins (on the 1st and 2nd) there will be a tailshot;
• the event consists in the fact that on the 1st coin the eagle falls and on the 2nd coin the tails;
• the event consists in the fact that on the 1st coin the tails fall and on the 2nd coin the eagle.

It is easy to see that the events are inconsistent (for example, two eagles can not fall out, and at the same time there are 2 tails) and form a complete group (since all possible outcomes of throwing two coins are taken into account ) . Let's sum up the event data: . How to interpret this record? Very simply - multiplication means logical connective AND , and addition - OR . Thus, the amount of easy to read understandable in human terms, "fall out two eagles , or two tails or the 1st coin falls an eagle , and on the 2nd tails , or on the 1st coin falls tails and on the 2nd coin of the eagle »

This was an example when in one test several objects are involved, in this case - two coins. Another common scheme in practical tasks is repeated tests , when, for example, the same playing cube is thrown three times in a row. As a demonstration, consider the following events:

• in the 1st throw will drop 4 points;
• in the 2nd throw will drop 5 points;
• in the third throw will drop 6 points.

Then the event is that in the 1st throw will drop 4 points and in the 2nd throw will drop 5 points and in the 3rd throw will drop 6 points. Obviously, in the case of a cube, there will be significantly more combinations (outcomes) than if we were tossing a coin.

... I understand that, perhaps, not very interesting examples are considered, but this is often found in the tasks of things and they can not escape from anywhere. In addition to a coin, a dice and a deck of cards, you are waiting for urns with colorful balls, several anonymous characters shooting at the target, and a tireless worker who constantly sharpens some details =)

Probability of the event
The probability of an event is the central concept of probability theory. ... A murderous logical thing, but with something it was necessary to start =) There are several approaches to its definition:

The classical definition of probability ;
Geometric definition of probability ;
Statistical definition of probability .

In this article I will focus on the classical definition of probabilities, which finds the most wide application in the study assignments.

Notation . The probability of an event is denoted by a large Latin letter , and the event itself is bracketed, acting as a kind of argument. For example:

• the probability that the "eagle" will drop out as a result of the roll of the coin;
• the likelihood that as a result of the dice roll will drop 5 points;
• the probability that a card of a club suit will be drawn from the deck.

A small letter is also used to denote probability . In particular, it is possible to abandon the cumbersome notation of events and their probabilities in favor of the following stylistics:

• the probability that the "eagle" will drop out as a result of the roll of the coin;
• the likelihood that as a result of the dice roll will drop 5 points;
• the probability that a card of a club suit will be drawn from the deck.

This option is popular in solving practical problems, because it allows you to significantly reduce the recording of the solution. As in the first case, it is convenient to use "speaking" subscript / superscript indices.

Everyone has already guessed the numbers that I just wrote above, and now we'll find out how they turned out:

The classical definition of probability :
The probability of occurrence of an event in a certain trial is the relation , where:

• the total number of all equally probable , elementary outcomes of this test, which form a complete group of events ;

• the number of elementary outcomes that favor the event .

When the coin is rolled, either the eagle or the tails can fall out - these events form a complete group , thus the total number of outcomes ; moreover, each of them is elementary and is equally possible . The event is favored by the outcome (falling out of the eagle). The classic definition of probabilities .

Similarly, as a result of throwing a dice, elementary equally probable outcomes may appear , forming a complete group, and the event is favored by a single outcome (loss of the five). Therefore: .

I draw particular attention to the third example. Here it will be incorrect to say "once in a pack of 4 suits, then the probability of extracting clubs " . In the definition, we are talking about elementary outcomes, so the correct order of reasoning is this: there are 36 cards in the deck (incompatible elementary outcomes forming a complete group) , of which 9 cards of a club suit (number of elementary outcomes that favor the event ) ; according to the classical definition of probability: . Exactly!

Probabilities can also be expressed in percentages, for example: the probability of an eagle falling out is equal to , the fall of a five , the extraction of a club , but in probability theory IT IS NOT ACCEPTED (although it is not forbidden to estimate interest in the mind).

It is customary to use the fractions of a unit , and it is obvious that the probability can vary within . In this case , if , then the event is impossible , if - authentic , and if , then it is a random event.

!! If in the course of solving any problem you have obtained some other probability value - look for an error!

In the classical approach to determining the probability, the extreme values ​​(zero and one) are obtained by exactly the same reasoning. Suppose that from a certain urn, in which there are 10 red balls, one ball is extracted at random. Consider the following events:

• a ball will be removed from the urn;
• A green ball will be removed from the urn.

The total number of outcomes: . The event is favored by all possible outcomes , therefore , that is, this event is reliable . For the second event , there are no favorable outcomes , therefore , that is, the event is impossible .

Of particular interest are events whose probability of occurrence is extremely small. Although such events are random, the following postulate is valid for them:

in a single trial, a small event will not happen .

That's why you will not break the Jackpot in the lottery if the probability of this event is, say, 0.00000001. Yes, yes, it is you - with a single ticket in a particular print run. However, a greater number of tickets and a greater number of drawings will not help you much . ... When I tell this to others, I almost always hear back: "But someone wins." Well, then let's conduct the following experiment: please, today or tomorrow, buy a ticket of any lottery (do not delay!). And if you win ... well, at least 10 kilobbles, be sure to unsubscribe - I'll explain why it happened. For a percentage, of course =) =)

But you do not need to be sad, because there is an opposite principle: if the probability of some event is very close to unity, then in a single trial it will practically reliably occur. Therefore, before the jump with a parachute, do not be afraid, on the contrary - smile! After all, absolutely inconceivable and fantastic circumstances must arise, so that both parachutes can not.

Although all this is a lyric, because depending on the content of the event the first principle can turn out to be fun, and the second - sad; or generally both parallel.

Perhaps, for the time being enough, in the lesson of the Problem on the classical definition of probability, we squeeze the maximum out of the formula . In the concluding part of this article, let us consider one important theorem:

The sum of the probabilities of events that form a complete group is one . Roughly speaking, if events form a complete group, then with 100% probability some of them will occur. In the simplest case, a complete group is formed by opposite events, for example:

• as a result of a roll of the coin, an eagle will fall out;
• as a result of a coin drop, a tailshot falls out.

By the theorem:

It is quite clear that these events are equally possible and their probabilities are the same .

Because of the equality of probabilities, equally likely events are often called equiprobable . And here's the patter for determining the degree of intoxication turned out =)

Example with a cube: the events are opposite, therefore .

This theorem is convenient in that it makes it possible to quickly find the probability of the opposite event. So, if the probability is known that a five will fall, it is easy to calculate the probability that it will not fall out:

This is much simpler than summarizing the probabilities of five elementary outcomes. For elementary outcomes, by the way, this theorem is also true:

Events , as noted above, are equally possible - and now we can say that they are equally probable. The probability of loss of any face of the cube is :

Well, on the snack deck: since we know the probability that the club will be extracted, it is easy to find the probability that a card of another suit will be drawn:

Note that the pair discussed the events and not equally as it often happens.

In the simplified version of the solution record, the probability of the opposite event is indicated by a lowercase letter . For example, if - the probability that the shooter hits the target, then - the probability that he misses.

!! In probability theory, the letters and undesirable to use in any other purposes.

In honor of Knowledge Day, I will not ask homework =), but it is very important that you can answer the following questions:

• What kinds of events exist?
• What is randomness and the equal possibility of an event?
• How do you understand the terms compatibility / inconsistency of events?
• What is a complete group of events, the opposite events?
• What does addition and multiplication mean?
• What is the essence of the classical definition of probability?
• How useful is the addition theorem for the probabilities of events forming a complete group?

No, there is nothing to cram one, it's just the basics of probability theory - a kind of primer that will fit pretty quickly into your head. And to make this happen as soon as possible, I suggest that you familiarize yourself with the lessons on Combinatorics and Problems on the classical definition of probability