1. Of the five letters of the alphabet is made cutting the word "book". A child who can not read, the letters scattered and then gathered them in a random order. Find the probability that he again turned the word "book".
2. The expert assesses the quality of the three types of products on consumer characteristics. On Probability that the product of the first type will be assigned a mark of quality is equal to 0.7; for the product of the second type, this probability is 0.9; and for the products of the third kind 0.8. Find the probability that the quality mark will be assigned to: a) all products; b) Only one product; c) at least one product
3. Three hunters shoot simultaneously and independently in the wild boar. It is known that the first flagged with probability 0.8, 0.4 second, and the third - 0.2. Boar killed, and there found two bullets. How to share a wild boar?
4. The Bank has six branches. With probability 0.2 independently of the other, each department can order the next day a large sum of money. At the end of the day one of the bank's vice president met with the received applications.
What is the probability that there will be: a) exactly two applications; b) at least one application. What is the probability that there is an application from the first compartment, if received two applications?
5. The probability of receiving each of the transmitted signal 100 is 0.75. Find the probability that there will be taken: a) exactly 70 signals; b) from 71 to 80 signals.
6. An investor can create a portfolio of various types of securities, the rate of profit which are random variables X], ..., Xn, MX1 = ai, DXi = σ, cov (Xi, XJ) = O, i ≠ j. To determine the proportion of capital investment in various securities, providing an average rate of a = 10, and minimizing the dispersion rate of return of the portfolio based on the following data:
7. In a group of 1,000 people 452 people have current accounts, 336 - have savings accounts and 302 - and current and deposit accounts. Are the attributes of "possession of a deposit account" and "possession of a current account" statistically independent?
8. Assuming that the relationship between the monthly advertising costs and the corresponding sales volumes has the form y = a + h0,5, where x - ad spending, while - sales. Consider the following table which shows the data on the volume of sales and advertising costs for the previous 12 months.
a) To put these values \u200b\u200bon a scatter plot. Explain the basis of the resulting graph, whether this dependence is linear or nonlinear.
b) Calculate the degree of correlation between x and y and comment on this dependence.
c) Determine the regression equation y from x index and use it to evaluate the volume of sales in any given month, provided that the advertising costs amount to £ 2000. Art.
d) Comment on the value obtained in part (c) and explain the reasons for which this forecast could be inaccurate.
9. The company, which produces tools for weight loss, claims that the pills in combination with a special diet allows you to lose an average of 400 g per week weight. Randomly selected 25 people who use this therapy, and found that the average weekly weight loss was 430 g with a standard deviation of 110 Test the hypothesis that the average weight loss of 400 g level of significance a = 0, 05.
10. The travel agency offers a place in the seaside resort hotels. Managers of the company are interested in how to increase the attractiveness of the hotel depending on the distance to the beach. For this purpose, the city has 14 hotels clarified the average occupancy rates and distance from the beach.
A. Distance, km
Plot the data and determine the source of his dependence. Calculate the sample linear correlation coefficient of Pearson, check its value at a = 0.05. Build a regression equation and give the interpre