USE SOLUTIONS IN MATHEMATICS - 2013
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Our system of testing and preparation for the exam I will SOLVE the Unified State Exam of the Russian Federation.

From 2001 to 2009, Russia began an experiment to combine school leaving exams with entrance exams to higher education institutions. In 2009, this experiment was completed, and since then a single State exam has become the main form of control over school preparation.

In 2010, the old team of exam writers was replaced by a new one. Together with the developers, the structure of the exam has also changed: the number of problems has decreased, the number of geometric problems has increased, and an Olympiad-type problem has appeared.

An important innovation was the preparation of an open bank of examination tasks, in which the developers posted about 75 thousand tasks. No one can solve this abyss of problems, but this is not necessary. In fact, the main types of tasks are represented by so-called prototypes, there are approximately 2400 of them. All other problems are obtained from them using computer cloning; they differ from prototypes only in specific numerical data.

Continuing, we present to your attention the solutions to all prototype exam tasks that exist in the open bank. After each prototype there is a list of clone tasks based on it for independent exercises.

Two factories produce the same glass for car headlights. The first factory produces 30% of these glasses, the second - 70%. The first factory produces 3% of defective glass, and the second - 4%. Find the probability that glass accidentally purchased in a store turns out to be defective.

Solution. Convert % to fractions.

Event A - "Glass from the first factory was purchased." P(A)=0.3

Event B - "Glass from the second factory was purchased." P(B)=0.7

Event X - "Defective glass".

P(A and X) = 0.3*0.03=0.009

P(B and X) = 0.7*0.04=0.028

According to the formula full probability:

P = 0.009+0.028 = 0.037

Answer: 0.037

Cowboy John hits a fly on the wall with a probability of 0.9 if he shoots from a zeroed revolver. If John fires an unfired revolver, he hits the fly with probability 0.2.

There are 10 revolvers on the table, only 4 of which have been shot. Cowboy John sees a fly on the wall, randomly grabs the first revolver he comes across and shoots the fly. Find the probability that John misses.

Solution.

The probability that the gun is sighted is 0.4, and that it is not is 0.6.

The probability of hitting a fly with a pistol, if it is aimed, is 0.4*0.9=0.36.

The probability of hitting a fly if the gun is not shot is 0.6*0.2=0.12.

Probability of hitting: 0.36+0.12=0.48.

Probability of miss P=1-0.48=0.52

During artillery fire the automatic system fires a shot at the target. If the target is not destroyed, the system fires a second shot. Shots are repeated until the target is destroyed. The probability of destroying a certain target with the first shot is 0.4, and with each subsequent shot it is 0.6. How many shots will be required to ensure that the probability of destroying the target is at least 0.98?

Solution. The probability of hitting a target is equal to the sum of the probabilities of hitting it on the first or second or... kth shot.

We will calculate the probability of destruction with the kth shot, setting the values ​​k=1,2,3... And summing up the obtained probabilities

k=1 P=0.4 S=0.4

k=2 P=0.6*0.6=0.36 - the first shot misses, the second the target is destroyed

S=0.4+0.36=0.76

k=3 P=0.6*0.4*0.6 = 0.144 - the target is destroyed on the third shot

S=0.76+0.144=0.904

k=4 P=0.6*0.4*0.4*0.6= 0.0576 - at 4th

S=0.904+0.0576=0.9616

k=5 P=0.6*0.4 3 *0.6 = 0.02304

S=0.9616+0.02304=0.98464 - reached the required probability at k=5.

Answer: 5.

To advance to the next round of competition, The football team needs to score at least 4 points in two games. If a team wins, it receives 3 points, in case of a draw - 1 point, if it loses - 0 points. Find the probability that the team will advance to the next round of the competition. Consider that in each game the probabilities of winning and losing are the same and equal to 0.4.

Solution. 4 points or more in two games can be scored in the following ways:

3+1 won, draw

1+3 draw, won

3+3 won both times

The probability of winning is 0.4, losing - 0.4, the probability of a draw is 1-0.4-0.4 = 0.2.

P = 0.4*0.2 + 0.2*0.4 + 0.4*0.4 = 2*0.08+0.16 = 0.32

Answer: 0.32

Try to decide for yourself:

In a batch of 800 bricks there are 14 defective ones. The boy selects one brick from this lot at random and throws it from the eighth floor of the construction site. What is the probability that a thrown brick will be defective?

The examination book in physics for grade 11 consists of 75 tickets. In 12 of them there is a question about lasers. What is the probability that Styopa's student, choosing a ticket at random, will come across a question about lasers?

At the 100m running championship there are 3 athletes from Italy, 5 athletes from Germany and 4 from Russia. The lane number for each athlete is determined by drawing lots. What is the probability that an athlete from Italy will be in the second lane?

At the Kievsky railway station in Moscow there are 28 ticket office windows, next to which 4,000 passengers are crowding, wanting to buy train tickets. Statistically, 1,680 of these passengers are inadequate. Find the probability that the cashier sitting at the 17th window will encounter an inadequate passenger (taking into account that passengers choose a ticket office at random).

In Vladivostok, a school was renovated and 1,200 new ones were installed plastic windows. An 11th grade student who did not want to take the Unified State Exam in mathematics found 45 cobblestones on the lawn and began throwing them at the windows at random. In the end, he broke 45 windows. Find the probability that the window in the director's office will not be broken.

Granny keeps it in her attic country house 2400 jars of cucumbers. It is known that 870 of them have long since gone rotten. When her granddaughter came to visit her, she gave him one jar from her collection, choosing it at random. What is the probability that your granddaughter received a jar of rotten cucumbers?

A team of 7 migrant construction workers offers apartment renovation services. During the summer season, they completed 360 orders, and in 234 cases they did not remove construction waste from the entrance. Utility services select one apartment at random and check the quality repair work. Find the probability that utility workers will not stumble upon construction waste when checking.

Condition

Cowboy John has a 0.9 chance of hitting a fly on the wall if he fires a zeroed revolver. If John fires an unfired revolver, he hits the fly with probability 0.2. There are 10 revolvers on the table, only 4 of which have been shot. Cowboy John sees a fly on the wall, randomly grabs the first revolver he comes across and shoots the fly. Find the probability that John misses.

Solution

Consider event A: “John will take the sighted revolver from the table and miss.” According to the conditional probability theorem (the probability of the product of two dependent events is equal to the product of the probability of one of them by the conditional probability of the other, found under the assumption that the first event has already occurred)

$=\frac(4)(10)\cdot (1-0.9)=0.04$,

where $=\frac(m)(n)=\frac(4)(10)$ is the probability of taking a sighted pistol from the table, and the probability of missing it (the opposite event of hitting the target) is equal to \

Consider event B: “John takes the unfired revolver from the table and misses.” Similar to the first one, let’s calculate the probability

$=\frac(10-4)(10)\cdot (1-0.2)=$0.48.

Events A and B are incompatible (cannot happen at the same time), which means that the probability of their sum is equal to the sum of the probabilities of these events:

Let's give another solution

John hits a fly if he grabs a zeroed revolver and shoots with it, or if he grabs an unshooted revolver and shoots with it. According to the conditional probability formula, the probabilities of these events are equal to \ and \, respectively. These events are incompatible, the probability of their sum is equal to the sum of the probabilities of these events: 0.36 + 0.12 = 0.48. The event that John misses is the opposite. Its probability is 1 − 0.48 = 0.52.

Cowboy John has a 0.7 chance of hitting a fly on the wall if he shoots with a zeroed revolver. If John fires an unfired revolver, then the probability that he will hit is 0.3. There are 10 revolvers on the table, only 2 of which have been shot. Cowboy John takes the first revolver he comes across and shoots a fly. Find the probability that John misses.

Solution

Cowboy John will only miss if one of the following events occurs:

• event A - cowboy John will miss with a aimed revolver
• event B - cowboy John will miss with an unfired revolver

Event A occurs when John grabs the shot revolver, i.e. the probability is equal \frac(2)(10) and if John misses with it, i.e. the probability is 1 − 0.7.

This means that the probability of event A occurring is:

P(A)=\frac(2)(10)\cdot(1-0.7)=\frac(2)(10)\cdot\frac(3)(10)=\frac(6)(100) =0.06

Event B occurs when John grabs the fired revolver, i.e. the probability is equal \frac(8)(10) and if John misses with it, i.e. the probability is 1 − 0.3.

This means that the probability of event B occurring is:

P(B)=\frac(8)(10)\cdot(1-0.3)=\frac(8)(10)\cdot\frac(7)(10)=\frac(56)(100) =0.56

John will miss if either event A or event B occurs, so the answer will be the sum of these events:

P=P(A)+P(B)=0.06+0.56=0.62