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ECON 210 - Economic Statistics Problem Set III Part I - Theoretical Probability and Combinatorics 1) Craps is a gambling game in which the players...

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ECON 210 - Economic Statistics Problem Set III Part I - Theoretical Probability and Combinatorics 1) Craps is a gambling game in which the players make a wager on the outcome of a roll of a a pair of dice. The basic rules of the game, whether it be played in a casino or on the street, are quite simple. Each player (called a shooter) gets turns rolling two dice, and calculating the value of the roll as the sum of the face values. Each turn has two phases: “come out” and “point”. In order to enter the game the shooter has to make a “come out” roll. A come out roll of value 2, 3, or 12 is called “craps” or “crapping out”, and the shooter loses the initial wager she made. A come out roll of 7 or 11 is called a “natural” and the shooter wins the bet automatically. If the roll lands on any other possible number, i.e. 4, 5, 6, 8, 9, 10 it establishes that value as the “point”. Once a point has been established, the shooter must roll the same point value again in order to win. If she rolls a 7 before she rolls her point, she is said to have “sevened out”, and loses the bet. Keeping in mind that the game involves rolling a pair of dice, answer the following questions. (i) How many events constitute the sample space of a roll of a pair of dice? (ii) What is the probability that a shooter will “crap out” on the come out roll? (iii) What is the probability that the shooter rolls a “natural”? (iv) What is, then, the probability of establishing a “point”? (Hint: Remind yourself of the very basic rules of probability we discussed and save yourself the trouble of doing too many calculations!) 2) Car number plates in California follow the pattern 1 ABC 234, i.e. the first character is a number (only from 1-9), the next three characters are letters and the last three are again numbers (from 0-9). (i) How many unique number plates can be created of this pattern? (ii) How many number plates can be created if no two characters on the plate were allowed to be repeated? 3) In a random arrangement of the letters of the word VIOLENT, find the probability that the vowels occupy the even spots. 4) An urn 1 has 8 black, 3 red and 9 white balls. If 3 balls were drawn from the urn at random, find the probability that: (i) all are black (ii) 2 are black and 1 is white, (iii) one is of each color, (iv) none are red. 5) A pound has 60 German Shepherds and 40 Great Danes of both sexes. If they breed with each other randomly, what is the probability that the puppies are: (i) pure breed German Shepherds, (ii) pure breed Great Danes, (iii) German Shepherd-Great Danes? ] 6) Five digit numbers are being formed from the numbers 1, 2, 3, 4, 5; no digit being repeated. Find the probability that a random number so formed is: (i) divisible by 5, (ii) divisible by 2, (iii) greater than 23,000. 1 An urn is a kind of a tall vase, usually used to store the ashes of a cremated person, but let’s not dwell on the morbid stuff. 1
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7) A mailman delivers 90 letters to an office, 50 of which are supposed to go to to the Accounting Department and 40 of which are supposed to go to the Marketing Department. However the mail- man finds that the exact address on 2 of these letters have been washed away by a sudden downpour which caught him on the way in to the office. Needless to say, a sudden downpour washing away addresses is a completely random event. The mailman takes a chance and delivers both letters to the Accounting Department. What is the probability that (i) he delivered both letters to the wrong department, (ii) he delivered both letters to the correct department, (iii) he delivered one letter right and one wrong? Part II - Laws of Probability, Conditional Probability and Baye’s Rule 8) You apply for a job at General Electric and Miller Brewing Company. You estimate your proba- bility of selection at General Electric to be 0.7 and being rejected at Miller is 0.5. The probability that at least one of your applications is rejected is 0.6. What is then the probability that you land a job? [Hint: Remember P (at least one) = 1 - P (none)] 9) A problem in statistics is given to two students A and B. The odds in favor of A solving the problem are 6 to 9 (or equivalently the “odds ratio” is 6/9) and the odds in favor of B solving the problem is 10 to 12. Assuming they work independently and both attempt the problem, find the probability of the problem being solved. [Hint: To find probability of any event A from odds you follow the formula: P ( A ) = Odds Ratio 1+Odds Ratio ] 10) The probability that a management trainee will remain with a firm is 0.6. The probability that an employee earns more than $70,000 per annum is 0.5. The probability that an employee is a management trainee who stayed with the firm or an employee earns more than $70,000 is 0.7. You want to appraise all employees who are both management trainees and who earn more than $70,000. (i) What is the probability that an employee is a management trainee and earning more than $70,000? (ii) You randomly select a management trainee without knowing her earnings. What is the proba- bility that she earns more than $70,000? [Hint: Think conditional probability] 11) You are down with a fever and have an ugly looking rash forming on your face. You, not want- ing to wait to go to a doctor, look up the symptoms on WebMD and find that they are consistent with having Lupus. In fact, you find, 95% of those who have Lupus show these symptoms. You start to panic, but you are suddenly reminded of a (heretofore useless) Economic Statistics class you took one summer that taught you Conditional Probability. Upon digging around a little more on Google, you find that the probability that any person contracts Lupus in the US in a given year is approximately 8 per 100,000. Also, you find that your symptoms are fairly common, around 1 in a 100 people can have it in a year. Armed with these numbers and the new-found knowledge that the class taught you how to figure this out; what is the probability that you have Lupus given that you have these symptoms? 12) A UK based betting company estimates that the probability that Donald Trump becomes the next US President is 0.25, and the corresponding probability for Hillary Clinton is 0.75. If Donald Trump wins, the probability that a gun control bill gets introduced is 0.3 and if Hillary Clinton wins the corresponding probability is 0.8. What is the probability that a gun control bill will be introduced in the next presidential term? 2
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1) i)12
ii)0.25 iii)0.17 iv)0.5
2) i)5040 ii)120
3) 0.3
4) i)0.15 ii)0.03 iii)0.064 iv)0.94
5) i)0.6
6) i)0.4
ii)0.5 iii)0.3
7) i)0.5
ii)0.5 iii)0.06
8) 0.48
9) 10/11

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