8. Which of the following statements must be true if , and(i) and are mutually exclusive. Explanation:(ii) and are independent. Explanation:(iii) and are collectively exhaustive. Explanation:(iv) Explanation:a) All are true but (i)b) *All are true but (ii)c) All are true but (iii)d) All are true but (iv)e) All are truef) None are true.TABLE 4-3251y0322 10/27/03 ECO251 QBA1 SECOND HOUR EXAMMarch 21, 2003 Name: ____KEY_____________ Social Security Number: _____________________Part I: (48 points) Do all the following: All questions are 2 points each except as marked. Exam is normedon 50 points including take-home. Please re-read, ‘Things that You should never do on an Exam or Anywhere Else, ‘ and especially recall that a probability cannot be above 1!The following joint probability table shows the relation between two sets of events. Let event A be that the individual is below 22 (A is over 21), and the event 0B be that the individual had no traffic violations in the last 18 months, the event 1B be that the individual has one traffic violation in the last 18 months and the event 2B be that the individual has 2 traffic violations over the last 18 months. No individuals in this pool of drivers has more than 2 violations.01.18.41.06.12.22.210AABBB Note, to do the problems below, at least total the rows and columns -00.107.30.63.60.40.01.18.41.06.12.22.210AABBB1. The probability that someone who is over 21 has no traffic violations is (To 2 decimal places):a) .63b) .60.c) .41d) *.68 You have been asked for 683.60.41.00APABPABPe) None of the above.2. The probability that someone picked at random is over 21 and has no traffic violations is (To 2 decimal places):a) .63b) .60.c) *.41 Joint probabilities are what the table shows!d) .68e) None of the above.3. The probability that someone chosen at random is either under 22 or has 2 violations is:a) .40b) .06c) .46d) .47e) *.41 41.06.07.40.222 BAPBPAPBAPf) None of the above1251y0322 10/27/0300.107.30.63.60.40.01.18.41.06.12.22.210AABBB4. Which two events are independent?a)A and ANote that ‘mutually exclusive’ and ‘independent are almost opposites. b)A and 2Bc) *A and 1B. The definition of independence is APBAP 1. In this case 60.AP and 60.30.18.111BPBAPBAP. But a better way to do this is to note that 18.30.60.11 BPAPBAP.d)A and 0Be)A and 2Bf) None of these.5. Which two events are mutually exclusive?a) *A and AComplements are always mutually exclusive. None of the other pairs have a joint probability of zero.b)A and 2Bc)A and 1B.d)A and 0Be)A and 2Bf) None of these.In questions 6 and 7 you need to know what 0BP, 1BP and 2BP are to do the problems. Showyour work. Solution: We can use the following table. 58.044.000.128.014.007.230.030.030.10063.02TotalxxxxPxxxPxPEvent6. What is the probability that a person picked at random has at least one violation?Solution: 37.07.30.211 PPxP ‘For the 200th time, ‘at least one’ and ‘exactly one’ are rarely the same thing.7. What is the mean and the standard deviation of the number of violations our drivers have? (6) 18Solution: From the work above 44.0xxP. 3864.44.58.2222xE6216.03864. This can’t be x and s. These are a sample mean and variance, but there is no sample.2251y0322 10/27/038. Which of the following statements must be true if 6.AP, 4.BP and 0 BAP(i) A and B are mutually exclusive. Explanation: 0 BAP(ii) A and B are independent. Explanation: BPAPBAP (iii) A and B are collectively exhaustive. Explanation: 1 BAP(iv) 0BAPExplanation: BPBAPBAPa) All are true but (i) b) *All are true but (ii) c) All are true but (iii) d) All are true but (iv)e) All are truef) None are true.9. According to a survey of American households, the probability that the residents own 2 cars if annual household income is over $25,000 is 80%. Of the households surveyed, 60% had incomes over$25,000 and 70% had 2 cars. The probability that the residents of a household own 2 cars and have an income less than or equal to $25,000 a year is (3):a) 0.12.b) 0.18.c) *0.22.d) 0.48.Solution: The easiest way I know is to start with the given facts. Let ‘two’ be the event that a household owns two cars, and ‘25’ be the event that a family has an income over $25000. The problem says 80.25 twoP, 60.25 P, 70.twoP and asks for 25twoPUsing the second two facts we get 00.1__60.70.________2525twotwo. But 80% of the 60% of families that own two cars have incomes over 25000, that is 252525 PtwoPtwoP .48.60.80. So now we have00.1__60.70.______48.2525twotwo If we fill in more, we get300.1__60.70.__12.22.48.2525twotwo and, finally,00.140.60.30.70.18.12.22.48.2525twotwo. From the table we read 22.25 t woP4251y0322 10/27/0310. According to a survey of American households, the probability that the residents own 2 cars if annual household income is over $25,000 is 80%. Of the households surveyed, 60% had incomes over$25,000 and 70% had 2 cars. The probability that annual household income is over $25,000 if the residents of a household own 2 cars is: (4)a) 0.42.b) 0.48.c) 0.50.d) 0.69. If we look at the table above 6857.70.48.2525 twoPtwoPtwoP. Orwe could use Bayes’ rule to say 6857.70.60.80.252525 twoPPtwoPtwoP11. If you toss a coin 6 times, what is the chance that you get at least one head? (Show your work) (4)Solution: The probability of at least one head is .984375.015625.15.116 TTTTTTP12. If you have five pennies and four dimes in your pocket and you pick three coins, what is the chance that you get exactly 2 dimes? (Show your work) (4) Solution: Remember that !!!rrnnCnr The number of ways you can get one penny from 5 is551C. The number of ways you can get two dimes from four is 61234!2!2!442C . The number of ways that
View Full Document
Unlocking...