Variant 3
1. A die is tossed once. Find the probabilities of the following events:
a) appearance of an odd number of aces;
b) appearance of no less than 4 aces;
c) appearance of no more than 2 aces.
2. Products of a certain kind are delivered in a shop from three factories. The first factory delivers 45% of all products, the second – 40%, and the third – 15%. The probability that a product of the first factory is non-defective is equal to 0,6, of the second – 0,7, and of the third – 0,85. One product has been bought. It appeared non-defective. Determine the probability that the bought product has been made by the third factory.
3. A coming up seeds of a given grade of cucumber is 0,8. Find the probability that the number of grown seeds will be between 775 and 810 from 1000 planted seeds.
4. Two independent random variables X and Y are given by the following tables of distribution:
X |
1 |
2 |
4 |
|
Y |
2 |
3 |
P |
0,4 |
0,3 |
0,3 |
|
P |
0,7 |
0,3 |
Compose the table of distribution of the random variable X – Y and check the property: D(X – Y) = D(X) + D(Y).
5.
A random variable X
is given by the density of distribution
in the interval (0; 4), and f(x)
= 0 outside of the interval. Find: a) the dispersion of the random
variable X;
b) the probability of hit of the random variable X
into the interval (2; 5).
6. A fat content of milk of cows in collective farms of a region (in percentage) is a normally distributed random variable with mathematical expectation 4 and mean square deviation 1. Find the probability that the random variable differs from its mathematical expectation no more than 0,3 (collective farm – колхоз).
7. The dispersion of each of pairwise independent random variables does not exceed 15. Estimate the probability that the deviation of the arithmetic mean of 2000 these variables from the arithmetic mean of their mathematical expectations will not exceed 0,5.
8. There are the following data on the size of annual revenues of 60 joint-stock companies:
812 |
521 |
1222 |
1140 |
1282 |
1244 |
698 |
1009 |
565 |
1396 |
723 |
701 |
755 |
592 |
1195 |
983 |
772 |
1125 |
638 |
1226 |
618 |
859 |
205 |
677 |
1086 |
1349 |
845 |
1250 |
732 |
1122 |
749 |
829 |
912 |
789 |
578 |
1310 |
926 |
1368 |
948 |
1085 |
632 |
918 |
545 |
872 |
909 |
1219 |
925 |
1294 |
608 |
720 |
502 |
1012 |
615 |
1228 |
1128 |
769 |
552 |
836 |
542 |
828 |
1) Compose the interval and the discrete variation series taking the beginning of the first interval equal 500, and the width of each interval equal 50.
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.