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# 8 problems in probability theory

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**04.09.2013**

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# Description

Target 1.2.

Twelve parking places arranged in one row. Parked randomly placed 8 cars. Describe:

a) the space of elementary events placing vehicles;

b) event A - "4 blank spaces follow one another";

c) In the event - "3 empty places follow one another."

Find the probability of events A and B. Are these events inconsistent and independent? Search.

Target 2.2.

On the interval (0, 1) random put two points. Let ξ and η coordinates of these points. It covers the following events:

A - "the second point is closer to the left end of the segment than the first point to the right";

B - "roots of reality";

With - ""; D - «».

Search:.

Task 2.27.

On the interval (0, 1) random put two points. Let ξ and η coordinates of these points. It covers the following events:

A - "the second point is closer to the left end of the segment than the first point to the right";

B - "roots of reality";

With - "";

D - «».

Search:.

Target 3.2.

The experiment is to toss three dice. Observed events:

A - "is not dropped more than 2 sixes";

B - "on the 2nd of bone fell out number 1";

C - "the sum of points on the 2nd and 3rd is even bones";

D - «amount of points in the 2nd and 1st odd bones."

1) The dependence or not the events A and B;

2) Calculate the probability of events;

3) Find the probability that the threefold repetition of the experiment will be at least 1 time in the event S.

Target 4.2.

In the fall assembly parts made three rifles. It is known that the first machine gives 0.4%, the second - the third and 0.2% - 0.6% of marriage. Find the probability of getting the assembly of defective parts, if the first machine received 500, the second - in 1000, and the third - 1,250 parts. If the item was defective, which of the three machines has made it likely?

Target 5.2.

a) How much you want to check the details to a probability of 0.9; 0.99; 0,999 can be expected that the absolute value of the frequency deviation from the probability suitable parts 0.9 that would be non-defective item not exceed 0.01 (in absolute value)?

b) At the faculty 730 students. The likelihood that the student will not come to class, is 0.1. Find the most probable number of students who do not attend the sessions and the probability of this event.

c) Considering the probability of a boy and a girl the same, find the probability that among infants 6 2 will be boys.

d) The probability that the product does not stand the test, equal to 0.001. Find the probability that in 5000 more than one product would not stand the test.

Target 6.2.

Create the distribution law of a discrete random variable ξ, calculate its expectation, variance, standard deviation, as well as draw a polygon of its distribution and a graph of the distribution function.

From an urn containing four white and 4 black balls randomly extracted three balls.

ξ - the number taken out of the black balls.

Target 7.2.

The random variable ξ given density function

Search for:

1) the distribution function F (x) and required constant;

2) the expectation, variance and standard deviation;

3) the probability of hitting the random variable ξ in the interval.

Build graphs of the distribution function F (x) and density distribution f (x).

Target 8.2.

The random variable ξ has a density distribution with a given variance.

1) Find the constants a and b;

2) Find the distribution function;

3) Determine whether or not the events are dependent and;

4) random variable ξ is measured in three independent trials. As a result of building a new random variable η, which is equal to 1 if at least one measuring event occurred A; is 0 if A happened not once, but at least once happened

B - A, and takes a value of -1 in all other cases. Locate and then.

# Additional information

The exact formulation of the tasks you can see in the attached image