How to find the probability of an event examples. Classical and statistical definition of probability

In economics, as in other areas of human activity or in nature, we constantly have to deal with events that cannot be accurately predicted. Thus, the sales volume of a product depends on demand, which can vary significantly, and on a number of other factors that are almost impossible to take into account. Therefore, when organizing production and carrying out sales, you have to predict the outcome of such activities on the basis of either your own previous experience, or similar experience of other people, or intuition, which to a large extent also relies on experimental data.

In order to somehow evaluate the event in question, it is necessary to take into account or specially organize the conditions in which this event is recorded.

The implementation of certain conditions or actions to identify the event in question is called experience or experiment.

The event is called random, if as a result of experience it may or may not occur.

The event is called reliable, if it necessarily appears as a result of a given experience, and impossible, if it cannot appear in this experience.

For example, snowfall in Moscow on November 30 is a random event. The daily sunrise can be considered a reliable event. Snowfall at the equator can be considered an impossible event.

One of the main tasks in probability theory is the task of determining a quantitative measure of the possibility of an event occurring.

Algebra of events

Events are called incompatible if they cannot be observed together in the same experience. Thus, the presence of two and three cars in one store for sale at the same time are two incompatible events.

Amount events is an event consisting of the occurrence of at least one of these events

An example of the sum of events is the presence of at least one of two products in the store.

The work events is an event consisting of the simultaneous occurrence of all these events

An event consisting of the appearance of two goods in a store at the same time is a product of events: - the appearance of one product, - the appearance of another product.

Events form a complete group of events if at least one of them is sure to occur in experience.

Example. The port has two berths for receiving ships. Three events can be considered: - the absence of ships at the berths, - the presence of one ship at one of the berths, - the presence of two ships at two berths. These three events form a complete group of events.

Opposite two unique possible events that form a complete group are called.

If one of the events that is opposite is denoted by , then the opposite event is usually denoted by .

Classical and statistical definitions of event probability

Each of the equally possible results of tests (experiments) is called an elementary outcome. They are usually designated by letters. For example, a die is thrown. There can be a total of six elementary outcomes based on the number of points on the sides.

From elementary outcomes you can create a more complex event. Thus, the event of an even number of points is determined by three outcomes: 2, 4, 6.

A quantitative measure of the possibility of the occurrence of the event in question is probability.

The most widely used definitions of the probability of an event are: classic And statistical.

The classical definition of probability is associated with the concept of a favorable outcome.

The outcome is called favorable to a given event if its occurrence entails the occurrence of this event.

In the above example, the event in question—an even number of points on the rolled side—has three favorable outcomes. In this case, the general
number of possible outcomes. This means that the classical definition of the probability of an event can be used here.

Classic definition equals the ratio of the number of favorable outcomes to the total number of possible outcomes

where is the probability of the event, is the number of outcomes favorable to the event, is the total number of possible outcomes.

In the considered example

The statistical definition of probability is associated with the concept of the relative frequency of occurrence of an event in experiments.

The relative frequency of occurrence of an event is calculated using the formula

where is the number of occurrences of an event in a series of experiments (tests).

Statistical definition. The probability of an event is the number around which the relative frequency stabilizes (sets) with an unlimited increase in the number of experiments.

In practical problems, the probability of an event is taken to be the relative frequency for a sufficiently large number of trials.

From these definitions of the probability of an event it is clear that the inequality is always satisfied

To determine the probability of an event based on formula (1.1), combinatorics formulas are often used, which are used to find the number of favorable outcomes and the total number of possible outcomes.

When a coin is tossed, you can say that it will land heads up, or probability this is 1/2. Of course, this does not mean that if a coin is tossed 10 times, it will necessarily land on heads 5 times. If the coin is "fair" and if it is tossed many times, then heads will land very close half the time. Thus, there are two types of probabilities: experimental And theoretical .

Experimental and theoretical probability

If we flip a coin a large number of times - say 1000 - and count how many times it lands on heads, we can determine the probability that it lands on heads. If heads are thrown 503 times, we can calculate the probability of it landing:
503/1000, or 0.503.

This experimental determination of probability. This definition of probability comes from observation and study of data and is quite common and very useful. Here, for example, are some probabilities that were determined experimentally:

1. The probability that a woman will develop breast cancer is 1/11.

2. If you kiss someone who has a cold, then the probability that you will also get a cold is 0.07.

3. A person who has just been released from prison has an 80% chance of returning to prison.

If we consider tossing a coin and taking into account that it is just as likely that it will come up heads or tails, we can calculate the probability of getting heads: 1/2. This is a theoretical definition of probability. Here are some other probabilities that have been determined theoretically using mathematics:

1. If there are 30 people in a room, the probability that two of them have the same birthday (excluding year) is 0.706.

2. During a trip, you meet someone, and during the conversation you discover that you have a mutual friend. Typical reaction: “This can’t be!” In fact, this phrase is not suitable, because the probability of such an event is quite high - just over 22%.

Thus, experimental probabilities are determined through observation and data collection. Theoretical probabilities are determined through mathematical reasoning. Examples of experimental and theoretical probabilities, such as those discussed above, and especially those that we do not expect, lead us to the importance of studying probability. You may ask, "What is true probability?" In fact, there is no such thing. Probabilities within certain limits can be determined experimentally. They may or may not coincide with the probabilities that we obtain theoretically. There are situations in which it is much easier to determine one type of probability than another. For example, it would be sufficient to find the probability of catching a cold using theoretical probability.

Calculation of experimental probabilities

Let us first consider the experimental definition of probability. The basic principle we use to calculate such probabilities is as follows.

Principle P (experimental)

If in an experiment in which n observations are made, a situation or event E occurs m times in n observations, then the experimental probability of the event is said to be P (E) = m/n.

Example 1 Sociological survey. An experimental study was conducted to determine the number of left-handed people, right-handed people and people whose both hands are equally developed. The results are shown in the graph.

a) Determine the probability that the person is right-handed.

b) Determine the probability that the person is left-handed.

c) Determine the probability that a person is equally fluent in both hands.

d) Most Professional Bowling Association tournaments are limited to 120 players. Based on the data from this experiment, how many players could be left-handed?

Solution

a)The number of people who are right-handed is 82, the number of left-handers is 17, and the number of those who are equally fluent in both hands is 1. The total number of observations is 100. Thus, the probability that a person is right-handed is P
P = 82/100, or 0.82, or 82%.

b) The probability that a person is left-handed is P, where
P = 17/100, or 0.17, or 17%.

c) The probability that a person is equally fluent in both hands is P, where
P = 1/100, or 0.01, or 1%.

d) 120 bowlers, and from (b) we can expect that 17% are left-handed. From here
17% of 120 = 0.17.120 = 20.4,
that is, we can expect about 20 players to be left-handed.

Example 2 Quality control . It is very important for a manufacturer to keep the quality of its products at a high level. In fact, companies hire quality control inspectors to ensure this process. The goal is to produce the minimum possible number of defective products. But since the company produces thousands of products every day, it cannot afford to test every product to determine whether it is defective or not. To find out what percentage of products are defective, the company tests far fewer products.
The USDA requires that 80% of the seeds sold by growers must germinate. To determine the quality of the seeds that an agricultural company produces, 500 seeds from those that were produced are planted. After this, it was calculated that 417 seeds sprouted.

a) What is the probability that the seed will germinate?

b) Do the seeds meet government standards?

Solution a) We know that out of 500 seeds that were planted, 417 sprouted. Probability of seed germination P, and
P = 417/500 = 0.834, or 83.4%.

b) Since the percentage of seeds germinated has exceeded 80% as required, the seeds meet government standards.

Example 3 Television ratings. According to statistics, there are 105,500,000 households with televisions in the United States. Every week, information about viewing programs is collected and processed. In one week, 7,815,000 households tuned in to the hit comedy series "Everybody Loves Raymond" on CBS and 8,302,000 households tuned in to the hit series "Law & Order" on NBC (Source: Nielsen Media Research). What is the probability that one household's TV is tuned to "Everybody Loves Raymond" during a given week? to "Law & Order"?

Solution The probability that the television in one household is tuned to "Everybody Loves Raymond" is P, and
P = 7,815,000/105,500,000 ≈ 0.074 ≈ 7.4%.
The chance that the household's TV was tuned to Law & Order is P, and
P = 8,302,000/105,500,000 ≈ 0.079 ≈ 7.9%.
These percentages are called ratings.

Theoretical probability

Suppose we are conducting an experiment, such as throwing a coin or darts, drawing a card from a deck, or testing products for quality on an assembly line. Each possible result of such an experiment is called Exodus . The set of all possible outcomes is called outcome space . Event it is a set of outcomes, that is, a subset of the space of outcomes.

Example 4 Throwing darts. Suppose that in a dart throwing experiment, a dart hits a target. Find each of the following:

b) Outcome space

Solution
a) The outcomes are: hitting black (B), hitting red (R) and hitting white (B).

b) The space of outcomes is (hitting black, hitting red, hitting white), which can be written simply as (H, K, B).

Example 5 Throwing dice. A die is a cube with six sides, each with one to six dots on it.

Suppose we are throwing a die. Find
a) Outcomes
b) Outcome space

Solution
a) Outcomes: 1, 2, 3, 4, 5, 6.
b) Outcome space (1, 2, 3, 4, 5, 6).

We denote the probability that an event E occurs as P(E). For example, “the coin will land on heads” can be denoted by H. Then P(H) represents the probability that the coin will land on heads. When all outcomes of an experiment have the same probability of occurring, they are said to be equally likely. To see the differences between events that are equally likely and events that are not, consider the target shown below.

For target A, the events of hitting black, red and white are equally probable, since the black, red and white sectors are the same. However, for target B, the zones with these colors are not the same, that is, hitting them is not equally probable.

Principle P (Theoretical)

If an event E can happen in m ways out of n possible equally probable outcomes from the outcome space S, then theoretical probability events, P(E) is
P(E) = m/n.

Example 6 What is the probability of rolling a die to get a 3?

Solution There are 6 equally probable outcomes on a dice and there is only one possibility of rolling the number 3. Then the probability P will be P(3) = 1/6.

Example 7 What is the probability of rolling an even number on a die?

Solution The event is the throwing of an even number. This can happen in 3 ways (if you roll a 2, 4 or 6). The number of equally probable outcomes is 6. Then the probability P(even) = 3/6, or 1/2.

We will use a number of examples involving a standard 52 card deck. This deck consists of the cards shown in the figure below.

Example 8 What is the probability of drawing an Ace from a well-shuffled deck of cards?

Solution There are 52 outcomes (the number of cards in the deck), they are equally likely (if the deck is well shuffled), and there are 4 ways to draw an Ace, so according to the P principle, the probability
P(draw an ace) = 4/52, or 1/13.

Example 9 Suppose we choose, without looking, one ball from a bag with 3 red balls and 4 green balls. What is the probability of choosing a red ball?

Solution There are 7 equally probable outcomes of drawing any ball, and since the number of ways to draw a red ball is 3, we get
P(red ball selection) = 3/7.

The following statements are results from Principle P.

Properties of Probability

a) If event E cannot happen, then P(E) = 0.
b) If event E is certain to happen then P(E) = 1.
c) The probability that event E will occur is a number from 0 to 1: 0 ≤ P(E) ≤ 1.

For example, in a coin toss, the event that the coin lands on its edge has zero probability. The probability that a coin is either heads or tails has a probability of 1.

Example 10 Let's assume that 2 cards are drawn from a 52-card deck. What is the probability that both of them are peaks?

Solution The number n of ways to draw 2 cards from a well-shuffled deck of 52 cards is 52 C 2 . Since 13 of the 52 cards are spades, the number of ways m to draw 2 spades is 13 C 2 . Then,
P(pulling 2 peaks)= m/n = 13 C 2 / 52 C 2 = 78/1326 = 1/17.

Example 11 Suppose 3 people are selected at random from a group of 6 men and 4 women. What is the probability that 1 man and 2 women will be selected?

Solution The number of ways to select three people from a group of 10 people is 10 C 3. One man can be chosen in 6 C 1 ways, and 2 women can be chosen in 4 C 2 ways. According to the fundamental principle of counting, the number of ways to choose 1 man and 2 women is 6 C 1. 4 C 2 . Then, the probability that 1 man and 2 women will be selected is
P = 6 C 1. 4 C 2 / 10 C 3 = 3/10.

Example 12 Throwing dice. What is the probability of rolling a total of 8 on two dice?

Solution Each dice has 6 possible outcomes. The outcomes are doubled, meaning there are 6.6 or 36 possible ways in which the numbers on the two dice can appear. (It’s better if the cubes are different, say one is red and the other is blue - this will help visualize the result.)

The pairs of numbers that add up to 8 are shown in the figure below. There are 5 possible ways to obtain a sum equal to 8, hence the probability is 5/36.

In the Unified State Examination tasks in mathematics, there are also more complex probability problems (than we considered in Part 1), where we have to apply the rule of addition, multiplication of probabilities, and distinguish between compatible and incompatible events.

So, the theory.

Joint and non-joint events

Events are called incompatible if the occurrence of one of them excludes the occurrence of others. That is, only one specific event or another can happen.

For example, when throwing a die, you can distinguish between events such as getting an even number of points and getting an odd number of points. These events are incompatible.

Events are called joint if the occurrence of one of them does not exclude the occurrence of the other.

For example, when throwing a die, you can distinguish such events as rolling an odd number of points and rolling a number of points that are a multiple of three. When a three is rolled, both events occur.

Sum of events

The sum (or combination) of several events is an event consisting of the occurrence of at least one of these events.

Wherein sum of two incompatible events is the sum of the probabilities of these events:

For example, the probability of getting 5 or 6 points on a die with one throw will be , because both events (rolling 5, rolling 6) are inconsistent and the probability of one or the other event occurring is calculated as follows:

The probability sum of two joint events equal to the sum of the probabilities of these events without taking into account their joint occurrence:

For example, in a shopping center two identical machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.3. The probability that both machines will run out of coffee is 0.12. Let’s find the probability that by the end of the day the coffee will run out in at least one of the machines (that is, either one, or the other, or both).

The probability of the first event “coffee will run out in the first machine” as well as the probability of the second event “coffee will run out in the second machine” according to the condition is 0.3. Events are collaborative.

The probability of the joint occurrence of the first two events according to the condition is 0.12.

This means that the probability that by the end of the day the coffee will run out in at least one of the machines is

Dependent and independent events

Two random events A and B are called independent if the occurrence of one of them does not change the probability of the occurrence of the other. Otherwise, events A and B are called dependent.

For example, when two dice are rolled simultaneously, one of them, say 1, and the other, 5, are independent events.

Product of probabilities

The product (or intersection) of several events is an event consisting of the joint occurrence of all these events.

If two occur independent events A and B with probabilities P(A) and P(B) respectively, then the probability of the occurrence of events A and B at the same time is equal to the product of the probabilities:

For example, we are interested in seeing a six appear on a die twice in a row. Both events are independent and the probability of each of them occurring separately is . The probability that both of these events will occur will be calculated using the above formula: .

See a selection of tasks to practice the topic.

Probability is the degree (relative measure, quantitative assessment) of the possibility of the occurrence of some event. When the reasons for some possible event to actually occur outweigh the opposite reasons, then this event is called probable, otherwise - unlikely or improbable. The preponderance of positive reasons over negative ones, and vice versa, can be to varying degrees, as a result of which the probability (and improbability) can be greater or lesser. Therefore, probability is often assessed at a qualitative level, especially in cases where a more or less accurate quantitative assessment is impossible or extremely difficult. Various gradations of “levels” of probability are possible.
The study of probability from a mathematical point of view constitutes a special discipline - probability theory. In probability theory and mathematical statistics, the concept of probability is formalized as a numerical characteristic of an event - a probability measure (or its value) - a measure on a set of events (subsets of a set of elementary events), taking values from
(\displaystyle 0)
(\displaystyle 1)
Meaning
(\displaystyle 1)
Corresponds to a reliable event. An impossible event has a probability of 0 (the converse is generally not always true). If the probability of an event occurring is
(\displaystyle p)
Then the probability of its non-occurrence is equal to
(\displaystyle 1-p)
In particular, the probability
(\displaystyle 1/2)
Means equal probability of occurrence and non-occurrence of an event.
The classical definition of probability is based on the concept of equal probability of outcomes. The probability is the ratio of the number of outcomes favorable for a given event to the total number of equally possible outcomes. For example, the probability of getting heads or tails in a random coin toss is 1/2 if it is assumed that only these two possibilities occur and that they are equally possible. This classical “definition” of probability can be generalized to the case of an infinite number of possible values - for example, if some event can occur with equal probability at any point (the number of points is infinite) of some limited region of space (plane), then the probability that it will occur in some part of this feasible region is equal to the ratio of the volume (area) of this part to the volume (area) of the region of all possible points.
The empirical “definition” of probability is related to the frequency of an event, based on the fact that with a sufficiently large number of trials, the frequency should tend to the objective degree of possibility of this event. In the modern presentation of probability theory, probability is defined axiomatically, as a special case of the abstract theory of set measure. However, the connecting link between the abstract measure and the probability, which expresses the degree of possibility of the occurrence of an event, is precisely the frequency of its observation.
The probabilistic description of certain phenomena has become widespread in modern science, in particular in econometrics, statistical physics of macroscopic (thermodynamic) systems, where even in the case of a classical deterministic description of the movement of particles, a deterministic description of the entire system of particles does not seem practically possible or appropriate. In quantum physics, the processes described are themselves probabilistic in nature.

What is probability?

The first time I encountered this term, I would not have understood what it was. Therefore, I will try to explain clearly.

Probability is the chance that the event we want will happen.

For example, you decided to go to a friend’s house, you remember the entrance and even the floor on which he lives. But I forgot the number and location of the apartment. And now you are standing on the staircase, and in front of you there are doors to choose from.

What is the chance (probability) that if you ring the first doorbell, your friend will answer the door for you? There are only apartments, and a friend lives only behind one of them. With an equal chance we can choose any door.

But what is this chance?

The door, the right door. Probability of guessing by ringing the first doorbell: . That is, one time out of three you will accurately guess.

We want to know, having called once, how often will we guess the door? Let's look at all the options:

You called 1st door
You called 2nd door
You called 3rd door

Now let’s look at all the options where a friend could be:

A. Behind 1st the door
b. Behind 2nd the door
V. Behind 3rd the door

Let's compare all the options in table form. A tick indicates options when your choice coincides with a friend's location, a cross - when it does not match.

How do you see everything Maybe options your friend's location and your choice of which door to ring.

A favorable outcomes for all . That is, you will guess once by ringing the doorbell once, i.e. .

This is probability - the ratio of a favorable outcome (when your choice coincides with your friend’s location) to the number of possible events.

The definition is the formula. Probability is usually denoted by p, therefore:

It is not very convenient to write such a formula, so we will take for - the number of favorable outcomes, and for - the total number of outcomes.

The probability can be written as a percentage; to do this, you need to multiply the resulting result by:

The word “outcomes” probably caught your eye. Since mathematicians call various actions (in our case, such an action is a doorbell) experiments, the result of such experiments is usually called the outcome.

Well, there are favorable and unfavorable outcomes.

Let's go back to our example. Let's say we rang one of the doors, but a stranger opened it for us. We didn't guess right. What is the probability that if we ring one of the remaining doors, our friend will open it for us?

If you thought that, then this is a mistake. Let's figure it out.

We have two doors left. So we have possible steps:

1) Call 1st door
2) Call 2nd door

The friend, despite all this, is definitely behind one of them (after all, he wasn’t behind the one we called):

a) Friend for 1st the door
b) Friend for 2nd the door

Let's draw the table again:

As you can see, there are only options, of which are favorable. That is, the probability is equal.

Why not?

The situation we considered is example of dependent events. The first event is the first doorbell, the second event is the second doorbell.

And they are called dependent because they influence the following actions. After all, if after the first ring the doorbell was answered by a friend, what would be the probability that he was behind one of the other two? Right, .

But if there are dependent events, then there must also be independent? That's right, they do happen.

A textbook example is tossing a coin.

Toss a coin once. What is the probability of getting heads, for example? That's right - because there are all the options (either heads or tails, we neglect the probability of the coin landing on its edge), but it only suits us.
But it came up heads. Okay, let's throw it again. What is the probability of getting heads now? Nothing has changed, everything is the same. How many options? Two. How many are we happy with? One.

And let it come up heads at least a thousand times in a row. The probability of getting heads at once will be the same. There are always options, and favorable ones.

It is easy to distinguish dependent events from independent ones:

If the experiment is carried out once (they throw a coin once, ring the doorbell once, etc.), then the events are always independent.
If an experiment is carried out several times (a coin is thrown once, the doorbell is rung several times), then the first event is always independent. And then, if the number of favorable ones or the number of all outcomes changes, then the events are dependent, and if not, they are independent.

Let's practice determining probability a little.

Example 1.

The coin is tossed twice. What is the probability of getting heads twice in a row?

Solution:

Let's consider all possible options:

Eagle-eagle
Heads-tails
Tails-Heads
Tails-tails

As you can see, there are only options. Of these we are only satisfied. That is, the probability:

If the condition simply asks you to find the probability, then the answer must be given in the form of a decimal fraction. If it were specified that the answer should be given as a percentage, then we would multiply by.

Answer:

Example 2.

In a box of chocolates, all the chocolates are packaged in the same wrapper. However, from sweets - with nuts, with cognac, with cherries, with caramel and with nougat.

What is the probability of taking one candy and getting a candy with nuts? Give your answer as a percentage.

Solution:

How many possible outcomes are there? .

That is, if you take one candy, it will be one of those available in the box.

How many favorable outcomes?

Because the box contains only chocolates with nuts.

Answer:

Example 3.

In a box of balloons. of which are white and black.

What is the probability of drawing a white ball?
We added more black balls to the box. What is now the probability of drawing a white ball?

Solution:

a) There are only balls in the box. Of them are white.

The probability is:

b) Now there are more balls in the box. And there are just as many whites left - .

Answer:

Total probability

The probability of all possible events is equal to ().

Let's say there are red and green balls in a box. What is the probability of drawing a red ball? Green ball? Red or green ball?

Probability of drawing a red ball

Green ball:

Red or green ball:

As you can see, the sum of all possible events is equal to (). Understanding this point will help you solve many problems.

Example 4.

There are markers in the box: green, red, blue, yellow, black.

What is the probability of drawing NOT a red marker?

Solution:

Let's count the number favorable outcomes.

NOT a red marker, that means green, blue, yellow or black.

Probability of all events. And the probability of events that we consider unfavorable (when we take out a red marker) is .

Thus, the probability of pulling out a NOT red felt-tip pen is .

Answer:

The probability that an event will not occur is equal to minus the probability that the event will occur.

The rule for multiplying the probabilities of independent events

You already know what independent events are.

What if you need to find the probability that two (or more) independent events will occur in a row?

Let's say we want to know what is the probability that if we flip a coin once, we will see heads twice?

We have already considered - .

What if we toss a coin once? What is the probability of seeing an eagle twice in a row?

Total possible options:

Eagle-eagle-eagle
Heads-heads-tails
Heads-tails-heads
Heads-tails-tails
Tails-heads-heads
Tails-heads-tails
Tails-tails-heads
Tails-tails-tails

I don’t know about you, but I made mistakes several times when compiling this list. Wow! And only option (the first) suits us.

For 5 throws, you can make a list of possible outcomes yourself. But mathematicians are not as hardworking as you.

Therefore, they first noticed and then proved that the probability of a certain sequence of independent events each time decreases by the probability of one event.

In other words,

Let's look at the example of the same ill-fated coin.

Probability of getting heads in a challenge? . Now we flip the coin once.

What is the probability of getting heads in a row?

This rule doesn't only work if we are asked to find the probability that the same event will happen several times in a row.

If we wanted to find the sequence TAILS-HEADS-TAILS for consecutive tosses, we would do the same.

The probability of landing heads is - , heads - .

Probability of getting the sequence TAILS-HEADS-TAILS-TAILS:

You can check it yourself by making a table.

The rule for adding the probabilities of incompatible events.

So stop! New definition.

Let's figure it out. Let's take our worn-out coin and toss it once.
Possible options:

Eagle-eagle-eagle
Heads-heads-tails
Heads-tails-heads
Heads-tails-tails
Tails-heads-heads
Tails-heads-tails
Tails-tails-heads
Tails-tails-tails

So, incompatible events are a certain, given sequence of events. - these are incompatible events.

If we want to determine what the probability of two (or more) incompatible events is, then we add the probabilities of these events.

You need to understand that heads or tails are two independent events.

If we want to determine the probability of a sequence (or any other) occurring, then we use the rule of multiplying probabilities.
What is the probability of getting heads on the first toss, and tails on the second and third tosses?

But if we want to know what is the probability of getting one of several sequences, for example, when heads comes up exactly once, i.e. options and, then we must add up the probabilities of these sequences.

Total options suit us.

We can get the same thing by adding up the probabilities of occurrence of each sequence:

Thus, we add probabilities when we want to determine the probability of certain, inconsistent, sequences of events.

There is a great rule to help you avoid getting confused when to multiply and when to add:

Let's go back to the example where we tossed a coin once and wanted to know the probability of seeing heads once.
What is going to happen?

Should fall out:
(heads AND tails AND tails) OR (tails AND heads AND tails) OR (tails AND tails AND heads).
This is how it turns out:

Let's look at a few examples.

Example 5.

There are pencils in the box. red, green, orange and yellow and black. What is the probability of drawing red or green pencils?

Solution:

What is going to happen? We have to pull (red OR green).

Now it’s clear, let’s add up the probabilities of these events:

Answer:

Example 6.

If a die is thrown twice, what is the probability of getting a total of 8?

Solution.

How can we get points?

(and) or (and) or (and) or (and) or (and).

The probability of getting one (any) face is .

We calculate the probability:

Answer:

Training.

I think now you understand when you need to calculate probabilities, when to add them, and when to multiply them. Is not it? Let's practice a little.

Tasks:

Let's take a card deck containing cards including spades, hearts, 13 clubs and 13 diamonds. From to Ace of each suit.

What is the probability of drawing clubs in a row (we put the first card pulled out back into the deck and shuffle it)?
What is the probability of drawing a black card (spades or clubs)?
What is the probability of drawing a picture (jack, queen, king or ace)?
What is the probability of drawing two pictures in a row (we remove the first card drawn from the deck)?
What is the probability, taking two cards, to collect a combination - (jack, queen or king) and an ace? The sequence in which the cards are drawn does not matter.

Answers:

In a deck of cards of each value, it means:
Events are dependent, since after the first card pulled out, the number of cards in the deck decreased (as did the number of “pictures”). There are total jacks, queens, kings and aces in the deck initially, which means the probability of drawing a “picture” with the first card:
Since we remove the first card from the deck, it means that there are already cards left in the deck, including pictures. Probability of drawing a picture with the second card:
Since we are interested in the situation when we take out a “picture” AND a “picture” from the deck, we need to multiply the probabilities:
Answer:
After the first card pulled out, the number of cards in the deck will decrease. Thus, two options suit us:
1) The first card is Ace, the second is Jack, Queen or King
2) We take out a jack, queen or king with the first card, and an ace with the second. (ace and (jack or queen or king)) or ((jack or queen or king) and ace). Don't forget about reducing the number of cards in the deck!

If you were able to solve all the problems yourself, then you are great! Now you will crack probability theory problems in the Unified State Exam like nuts!

PROBABILITY THEORY. AVERAGE LEVEL

Let's look at an example. Let's say we throw a die. What kind of bone is this, do you know? This is what they call a cube with numbers on its faces. How many faces, so many numbers: from to how many? Before.

So we roll the dice and we want it to come up or. And we get it.

In probability theory they say what happened auspicious event(not to be confused with prosperous).

If it happened, the event would also be favorable. In total, only two favorable events can happen.

How many are unfavorable? Since there are total possible events, it means that the unfavorable ones are events (this is if or falls out).

Definition:

Probability is the ratio of the number of favorable events to the number of all possible events. That is, probability shows what proportion of all possible events are favorable.

They denote probability with a Latin letter (apparently from the English word probability - probability).

It is customary to measure probability as a percentage (see topics and). To do this, the probability value must be multiplied by. In the dice example, probability.

And in percentage: .

Examples (decide for yourself):

What is the probability of getting heads when tossing a coin? What is the probability of landing heads?
What is the probability of getting an even number when throwing a die? Which one is odd?
In a box of simple, blue and red pencils. We draw one pencil at random. What is the probability of getting a simple one?

Solutions:

How many options are there? Heads and tails - just two. How many of them are favorable? Only one is an eagle. So the probability
It's the same with tails: .
Total options: (how many sides the cube has, so many different options). Favorable ones: (these are all even numbers:).
Probability. Of course, it’s the same with odd numbers.
Total: . Favorable: . Probability: .

Total probability

All pencils in the box are green. What is the probability of drawing a red pencil? There are no chances: probability (after all, favorable events -).

Such an event is called impossible.

What is the probability of drawing a green pencil? There are exactly the same number of favorable events as there are total events (all events are favorable). So the probability is equal to or.

Such an event is called reliable.

If a box contains green and red pencils, what is the probability of drawing green or red? Yet again. Let's note this: the probability of pulling out green is equal, and red is equal.

In sum, these probabilities are exactly equal. That is, the sum of the probabilities of all possible events is equal to or.

Example:

In a box of pencils, among them are blue, red, green, plain, yellow, and the rest are orange. What is the probability of not drawing green?

Solution:

We remember that all probabilities add up. And the probability of getting green is equal. This means that the probability of not drawing green is equal.

Remember this trick: the probability that an event will not occur is equal to minus the probability that the event will occur.

Independent events and the multiplication rule

You flip a coin once and want it to come up heads both times. What is the likelihood of this?

Let's go through all the possible options and determine how many there are:

Heads-Heads, Tails-Heads, Heads-Tails, Tails-Tails. What else?

Total options. Of these, only one suits us: Eagle-Eagle. In total, the probability is equal.

Fine. Now let's flip a coin once. Do the math yourself. Happened? (answer).

You may have noticed that with the addition of each subsequent throw, the probability decreases by half. The general rule is called multiplication rule:

The probabilities of independent events change.

What are independent events? Everything is logical: these are those that do not depend on each other. For example, when we throw a coin several times, each time a new throw is made, the result of which does not depend on all previous throws. We can just as easily throw two different coins at the same time.

More examples:

The dice are thrown twice. What is the probability of getting it both times?
The coin is tossed once. What is the probability that it will come up heads the first time, and then tails twice?
The player rolls two dice. What is the probability that the sum of the numbers on them will be equal?

Answers:

The events are independent, which means the multiplication rule works: .
The probability of heads is equal. The probability of tails is the same. Multiply:
12 can only be obtained if two -ki are rolled: .

Incompatible events and the addition rule

Events that complement each other to the point of full probability are called incompatible. As the name suggests, they cannot happen simultaneously. For example, if we flip a coin, it can come up either heads or tails.

Example.

In a box of pencils, among them are blue, red, green, plain, yellow, and the rest are orange. What is the probability of drawing green or red?

Solution .

The probability of drawing a green pencil is equal. Red - .

Favorable events in all: green + red. This means that the probability of drawing green or red is equal.

The same probability can be represented in this form: .

This is the addition rule: the probabilities of incompatible events add up.

Mixed type problems

Example.

The coin is tossed twice. What is the probability that the results of the rolls will be different?

Solution .

This means that if the first result is heads, the second must be tails, and vice versa. It turns out that there are two pairs of independent events, and these pairs are incompatible with each other. How not to get confused about where to multiply and where to add.

There is a simple rule for such situations. Try to describe what is going to happen using the conjunctions “AND” or “OR”. For example, in this case:

It should come up (heads and tails) or (tails and heads).

Where there is a conjunction “and” there will be multiplication, and where there is “or” there will be addition:

Try it yourself:

What is the probability that if a coin is tossed twice, the coin will land on the same side both times?
The dice are thrown twice. What is the probability of getting a total of points?

Solutions:

(Heads fell and tails fell) or (tails fell and tails fell): .
What are the options? And. Then:
Dropped (and) or (and) or (and): .

Another example:

Toss a coin once. What is the probability that heads will appear at least once?

Solution:

Oh, how I don’t want to go through the options... Heads-tails-tails, Eagle-heads-tails,... But there’s no need! Let's remember about total probability. Do you remember? What is the probability that the eagle will never fall out? It’s simple: heads fly all the time, that’s why.

PROBABILITY THEORY. BRIEFLY ABOUT THE MAIN THINGS

Probability is the ratio of the number of favorable events to the number of all possible events.

Independent events

Two events are independent if the occurrence of one does not change the probability of the other occurring.

Total probability

The probability of all possible events is equal to ().

The probability that an event will not occur is equal to minus the probability that the event will occur.

The rule for multiplying the probabilities of independent events

The probability of a certain sequence of independent events is equal to the product of the probabilities of each event

Incompatible events

Incompatible events are those that cannot possibly occur simultaneously as a result of an experiment. A number of incompatible events form a complete group of events.

The probabilities of incompatible events add up.

Having described what should happen using the conjunctions “AND” or “OR”, instead of “AND” we put a multiplication sign, and instead of “OR” we put an addition sign.

Become a YouClever student,

Prepare for the Unified State Exam or Unified State Exam in mathematics,

And also get access to the YouClever textbook without restrictions...