Variant 23
1. There are 11 green and 7 red balls in a box. Two balls are taken out at random. What is the probability that these balls will be the same color?
2. A warehouse has received a big batch of goods from two factories, and 70% of them have been delivered from the first factory. It is known that 75% goods from the first factory and 65% goods from the second have the higher grade. What is the probability that a randomly taken unit of goods in the warehouse has the higher grade?
3. The probability of hit in a target at one shot by a shooter is equal to 0,8. Determine the most probable number of hits in the target at 25 shots and calculate the probability of this number.
4. Two independent random variables are given by the following tables of distribution:
X |
–1 |
0 |
2 |
|
Y |
1 |
2 |
P |
0,4 |
0,3 |
0,3 |
|
P |
0,4 |
0,6 |
Compose the law of distribution of the random variable X Y and check the property: M(X Y) = M(X) M(Y).
5.
A random variable X
has the density of distribution
in the interval (0; 1), and f(x)
= 0 outside of the interval. Find:
a) the parameter C and the distribution function F(x);
b)
the probability that the following inequality holds:
.
6. The height of an adult man is a random variable distributed under a normal law with parameters: a = 175 cm, = 6 cm. Find the density of probability and the probability that the height of a randomly chosen man is in the interval from 170 up to 180 cm.
7. The dispersion of each of 4000 independent random variables does not exceed 16. Estimate the probability that the absolute value of the deviation of the arithmetic mean of these variables from the arithmetic mean of their mathematical expectations will not exceed 0,8.
8. The following data on lifetime (in hours) of a separate vacuum cleaner have been obtained at inspecting 100 vacuum cleaners of a given set:
548 |
956 |
521 |
1220 |
575 |
1121 |
1039 |
989 |
223 |
403 |
630 |
680 |
605 |
302 |
759 |
1158 |
335 |
985 |
556 |
1079 |
889 |
856 |
969 |
749 |
798 |
901 |
1133 |
1010 |
581 |
670 |
525 |
500 |
91 |
1205 |
1095 |
603 |
567 |
279 |
631 |
374 |
799 |
585 |
1209 |
887 |
268 |
735 |
267 |
833 |
960 |
332 |
967 |
650 |
86 |
264 |
807 |
159 |
78 |
573 |
1075 |
875 |
631 |
995 |
135 |
1158 |
432 |
580 |
370 |
930 |
479 |
1067 |
558 |
1095 |
985 |
841 |
897 |
940 |
901 |
376 |
570 |
970 |
196 |
470 |
609 |
890 |
469 |
889 |
373 |
459 |
1195 |
611 |
699 |
611 |
339 |
154 |
875 |
988 |
1169 |
531 |
957 |
551 |
1) Compose the interval and the discrete variation series taking the beginning of the first interval equal 0, and the width of each interval equal 200.
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.