Variant 6

1. There are 100 tickets in a lottery, and 10 of them are winning. A participant of the lottery buys two tickets. Determine the probability that at least one of them will be winning.

2. At a factory making toys the first machine produces 60 %, the second – 30 %, and the third – 10 % of all toys. The probability that a randomly chosen toy produced by the first machine is defective equals 0,07, by the second – 0,03, and by the third – 0,01. What is the probability that a randomly chosen toy will be defective?

3. The probability that a coin lands on heads at one tossing is equal to 0,5. Calculate the probability that the coin lands on heads exactly two times at four tossings.

4. Two independent random variables X and Y are given by the following tables of distribution:

X	2	3		Y	0	2	3
P	0,6	0,4		P	0,2	0,6	0,2

Compose the table of distribution of the random variable Z = X + Y and check the property: M (X + Y) = M(X) + M (Y).

5. A random variable X is given by the integral function:

Find a) the differential function f(x); b) the mathematical expectation and the dispersion of X; c) the probability of hit of X into the interval (3/5; 6/5).

6. It is supposed that the durability limit of a let out party of a parachute fabric is a normally distributed random variable X with the dispersion equal 100 kg/cm². What maximal deviation of the durability limit of a tested sample of a parachute fabric from the mathematical expectation can be guaranteed with probability 0,925 by absolute value?

7. The probability of appearance of an event is 1/6 in each trial. By using the Chebyshev’s inequality estimate the probability that the number of appearances of the event differs from the mathematical expectation less than on 30 if 900 trials will be made.

8. There are the following data on the size of monthly salaries of bookkeepers of 100 commercial firms:

359	365	343	352	367	354	399	351	352	356
371	353	341	361	377	358	345	368	355	347
365	363	374	371	363	354	352	388	362	337
381	361	358	351	352	363	371	364	368	387
353	386	347	369	347	366	361	363	372	353
362	335	358	338	357	352	339	349	349	348
354	375	353	349	362	351	333	372	355	351
372	343	378	347	352	354	338	378	364	361
364	353	357	375	349	346	352	339	356	352
343	346	356	357	384	375	366	392	385	362

1) Compose the interval and the discrete variation series taking the beginning of the first interval equal 330, and the width of each interval equal 10.

2) Construct the histogram and the polygon of relative frequencies of distribution.

3) Find the mode and the median (using the discrete series).

4) Find empirical functions of distribution of continuous and discrete variation series; and construct their graphs.