Extra Practice Q

pdf

School

The University of Hong Kong *

*We aren’t endorsed by this school

Course

2901

Subject

Statistics

Date

Nov 24, 2024

Type

pdf

Pages

Uploaded by KidNeutron12719

The University of Hong Kong Department of Statistics and Actuarial Sciences STAT2901 Probability and Statistics: Foundations of Actuarial Science (2018-2019) Extra Practice Questions 1. In how many ways can 8 people be seated in a row if (a) there are no restrictions on the seating arrangement; (b) persons A and B must sit next to each other; (c) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other; (d) there are 5 men and they must sit next to each other; (e) there are 4 married couples and each couple must sit together? 2. The casino game sic-bo (also known as tai-sai, big-small) is a casino game played with three dice. How many different outcomes are there if (a) the three rolled dice are of different color (i.e. distinguishable); (b) the three rolled dice are of same color (i.e. indistinguishable). 3. A total of 7 different gifts are to be distributed among 10 children. (a) How many distinct results are possible if no child is to receive more than one gift? (b) If the 7 gift are all the same, what would be the answer to the question in (a)? 4. We have 20 thousand dollars that must be invested among 4 possible opportunities. Each investment must be integral in units of 1 thousand dollars, and there are minimal investments that need to be made if one is to invest in these opportunities. The minimal investments are 2, 2, 3, and 4 thousand dollars. How many different investment strategies are available if (a) an investment must be made in each opportunity; (b) investments must be made in at least 3 of the 4 opportunities? 5. How many different words can be formed by using three A letters, two B letters and one C letter? (The words may be meaningless.) 6. There are 4 identical red balls, 3 identical yellow balls, 2 identical blue balls and 1 white ball. In how many distinguishable ways can these 10 balls be arranged along a circle?

7. A box contains 100 balls, of which r are red. Suppose that the balls are drawn from the box one at a time, at random, without replacement. Determine (a) the probability that the first ball drawn will be red; (b) the probability that the 50 th ball drawn will be red; (c) the probability that the last ball drawn will be red. 8. If 8 castles are randomly placed on a chessboard, compute the probability that none of the castles can capture any of the others, i.e. the probability that no row or column contains more than one castle. 9. A box contains 24 light bulbs of which four are defective. If one person selects 10 bulbs from the box in a random manner, and a second person then takes the remaining 14 bulbs, what is the probability that all four defective bulbs will be obtained by the same person? 10. A deck of 52 cards contain 12 picture cards. If the 52 cards are distributed in a random manner among four players in such a way that each player receives 13 cards, what is the probability that each player will receive three picture cards? 11. Suppose that A and B are mutually exclusive events for which ( ) 3 . 0 = A P and ( ) 5 . 0 = B P . What are the probabilities that (a) either A or B occurs; (b) both A and B occur; (c) A occurs but B does not? 12. If 4 married couples are arranged in a row, find the probability that no husband sits next to his wife. 13. A certain town of population size 100,000 has 3 newspapers: I, II, and III. The proportion of townspeople that read these papers are as follows: I: 10% I and II: 8% I and II and III: 1% II: 30% I and III: 2% III: 5% II and III: 4% (The list tells us, for instance, that 8000 people read newspapers I and II.) (a) Find the number of people reading only one newspaper. (b) How many people read at least two newspapers? (c) If I and III are morning papers and II is an evening paper, how many people read at least one morning paper plus and evening paper? (d) How many people read only one morning paper and one evening paper?

14. The chance that a factory’s sprinkler system will fail is 20%; the chance that its alarm system will fail is 10%; and the chance that both will fail is 4%. After examining the sprinkler system, it is found to be still working. What is the chance that the alarm system is also working? 15. In a simplified version of bridge, two players A and B each draw one card in turn from a deck, and compare results. B wins the round if he or she has drawn a spade and A has not; if both have drawn spades, the round is a tie; otherwise, B loses the round. Find the probability that (a) B wins. (b) The round is a tie. (c) B loses. 16. A total of 48 percent of the women and 37 percent of the men that took a certain “quit smoking” class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of a year. If 62 percent of the original class were male, (a) what percentage of the original class attended the party? (b) what percentage of those attending the party were women? 17. A small plane has gone down, and the search is organized into three regions. Starting with the likeliest, they are: Region Initial Chance Plane is There Chance of Being Overlooked in the Search Mountains 0.5 0.3 Prairie 0.3 0.2 Sea 0.2 0.9 The last column gives the chance that if the plane is there, it will not be found. For example, if it went down at sea, there is 90% chance it will have disappeared, or otherwise not be found. Since the pilot is not equipped to long survive a crash in the mountains, it is particularly important to determine the chance that the plane went down in the mountains. (a) Before any search is started, what is this chance? (b) The initial search was in the mountains, and the plane was not found. Now what is the chance the plane is nevertheless in the mountains? (c) The search was continued over the other two regions, and unfortunately the plane was not found anywhere. Finally now what is the chance that the plane is in the mountains? (d) Describing how and why the chances changed from (a) to (b) to (c).

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

18. In a bag there are 4 red balls, 6 green balls, 6 red cubes. An object is randomly drawn from this bag. (a) What is the probability that a ball was drawn? What would you bet as the shape of the object, ball or cube? (b) Suppose when the object was drawn you found that it was red by a glimpse. What would you bet on now? (c) How many green cubes should be added to the bag so that knowing the color would given no information on the shape of the object drawn? 19. Prove or give counterexamples to the following statements: (a) If E and F are mutually exclusive, then they are independent. (b) If E is independent of F , and F is independent of G , then E is independent of G . (c) If E is independent of F , and E is independent of G , then E is independent of G F ∪ . (d) If E is independent of F , and E is independent of G , and φ = ∩ G F , then E is independent of G F ∪ . (e) Let G be an arbitrary event with ( ) 0 > G P . If E is independent of F , then they are conditionally independent given that G occurred. 20. There are two coins with identical shape, size and weight. One of these two coins is fair while the other one is biased with a probability of 0.8 to land on head if it is tossed. In order to determine which one is fair, a coin is randomly selected and tossed for 10 times. Suppose 7 heads were obtained. (a) Based on the observed experimental result, what is the probability that the selected coin is fair? (b) If in addition the other coin is also tossed for 10 times and 9 heads were obtained, what is the probability that the first tossed coin is fair? 21. Suppose that the probability of suffering a side effect from a certain flu vaccine is 0.005. If 1000 persons are inoculated, find approximately the probability that (a) at most one person suffers; (b) exactly four people suffer. 22. Suppose that the total number of scores in a soccer match of the England Premier League follows the Poisson distribution with 9 . 2 = λ , i.e. on average there are 2.9 goals per match. Determine (a) the probability that there will be more than 2 goals; (b) the probability that there will be even number of goals (zero is counted as even).

23. Two balls are chosen randomly from an urn containing 6 white, 3 black, and 1 orange balls. Suppose that we win $1 for each white ball drawn and we lose $1 for each orange ball drawn. Denote $ X as the amount that we can win. (a) What are the possible values of X ? (b) Determine the probability mass function and cumulative distribution function of X . 24. Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter 3 = λ . (a) Find the probability that 3 or more accidents occur today. (b) Repeat part (a) if it is known that at least 1 accident occurred today. 25. The probability that a machine produces a defective item is 0.01. Each item is checked as it is produced. Assume that these are independent trials. Compute the probability that at least 100 items must be checked to find one that is defective. 26. In the game of international bridge, a deck of 52 poker cards is randomly and evenly dealt to four players sitting at the four directions of the table: East, South, West, North. (a) What is the probability that the North will get exactly 3 Aces? (b) What is the probability that the South will get more than four spade cards?

Extra Practice Q

Related Documents