IWS chudesenko

1. Two dices tossed. Find probability that

A) the sum of point less than N

b) The product of point number less than N

C)The product of all points devided by N

2. There are four varieties of products, and the number of products of first grade is 1, second grade - 2, third grade - 3, fourth grade - 4. In order to control was taken at random m products. Determine the probability that one of them first-classis m1, andm2, m3,m4 second-class and third-class, forth-class respectively.

3. Among then lottery tickets have k winning . Find the probability that them selected at random will be l winning tickets.

4. In the elevator k-storey boarded n passengers (n <k). Each of them independently from others with the same probability can with draw at any (starting with the second) floor. Determine the probability that:

  1. All the passengers went on different floors;

2. Two passenger went on one floor.

5. In the section of unit length appears the point randomly. Determine the probability that the distance from a point to both ends of the section exceeds a value of 1 / k.

6. Moments of the beginning of the two events are distributed in the time interval from T1 to T2randomly. One event lasts 10 minutes, another - t min. Determine the probability that a) the events of the "overlapped" in time, and b) "non-overlapped."

7. In a circle of radius r, at random one point appears. Find the probability that ,it falls into one of the two disjoint figures, which have the areas equal to and ?

8. In two batches and % ​​of benign products, respectively. At random choose one item from each batch. What is the probability of finding among them: a) at least one defective, b) two defective, c) one benign and one defective

9. Probability of that goal hit with one shot the of first arrow is-, the second is-. First arrow did - shots, the second did - shots. Find the probability that the target is not hit.

10. Two players A and B alternately toss a coin. The winner is the one to whom fall the first emblem. The first player throw A, the second - B, third - A, etc.

1. Find the probability that A win sink-th throw.

2. What is the probability of winning of each player for very long game?

11. The urn contains M indexed balls numbered from 1 to M. The balls are removed one at a time without replacement. We consider the following events:

A – number of balls in the the order received gives the sequence 1, 2, ..., M;

B – at least once coincides number and serial number of the a ball recovered;

C – There are no matches ball numbers and the serial number received.

Determine the probability of events A, B, C. Find the probability for M tending to infinity.

12. From 1,000 lamps belong to the i-th batch, i=1,2,3, sum of ni=1000. In the first game– 6%, in the second– 5%, in the third– 4% are defective lamps. One lamp selected randomly. Determine the probability that the chosen lamp is defective.

13. N1 in the first urn of white and black balls of M1, the second N2 and M2 black. from the first to the second shifted K balls, then from the second litter removed one ball. Determine the probability that the selected ball – white

14. An album of pure k clearing and l brands are extracted m stamps (some of which may be clean and hydrated) are subject to special cancellation n return to the album. Then again extracted n marks. Determine the probability that all n brands clean

15. The store receives the same type of product from the three plants plant supplies 1 m (1,2,3) percent of the product of the plant m n (1,2,3) top-notch. bought one item. It was top-notch. Determine the likelihood that the purchased product produced by the J.

16. The coin thrown until tails do not fall out n times. Determine the probability that the figure drops m times.

17. The probability of winning in lottery is p on a ticket. n tickets are bought. Find the most probable number of winning tickets and the corresponding probability.

18. The probability that for each lottery ticket stand the large win is p1, p2 refers to the small win, and the probability of fail is. The n tickets are bought. Determine the probability of getting n 1 large win and n2 small.

19. Probability of failure of the telephone system at each call is p. n calls have been received. Determine the probability of m failure.

20. The probability of occurrence of an event in each of the n independent trials is p. Determine the probability that the event number m satisfies the following inequality

Variants 1-11

Variants 12-21

Variants 22-31

21. Given the density of the distribution p(x) of the random variable ζ. Find the parameter γ, the mathematical expectation Mζ, the dispersion Dζ, the distribution function of a random variable ζ, the probability of performance of the inequality <ζ<

Variants 1-8 p(x)=

Variants 9-16 p(x)=

Variants 17-24 p(x)=

22. The probabilityof density distribution ofa random variable ζ, is . Find the parameter γ, the mathematical expectation Mζ, the dispersion Dζ, the distribution function of a random variable ζ, the probability of performance of the inequality <ζ<.

23. By using the given law of distribution of a random variable, find the function the mathematical expectation Mζ, the dispersion Dζ of a random variable ζ.

Variants 1-10. Binomial law

P(ζ=k)=

Variants 11-20. Pascal’s law

P(ζ=k)=

Variants 21-31. Poisson’s law

P(ζ=k)=

24. Knowing the distribution law of a random variable ζ, find the characteristic formula and in variants 1-20 the mathematical expectation Mζ, the dispersion Dζ of a random variable ζ.

Variants 1-10. Random variable ζ uniformly distributed on the interval [a,b].

Variants 11-20. Random variable ζ has the distribution density of

P(x)=

21-24 tasks have been translated by Assylbekova Ainur (addition)