uiz - II 1. Find the sample space that describes all two-child families according to the genders of the children with respect to birth order. (a) {b, g} (b) {bb, gb} (c) {bb, gb, bg, gg} (d) {bb, gb, bg} 2. Construct a sample space for a random experiment of tossing of three coins. (a) {HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T } (b) {HH, HT, HT, T T } (c) {HHH, HHH, HT T, HT T, T HH, T HT, T T T, T T T } (d) {HT T, HT H, HT H, HT T, T HH, T HT, T T H, T T T } 3. What is the total number of possible outcomes when a die is rolled twice? (a) 26 (b) 36 (c) 32 (d) 28 4. In a sample space for the experiment that consists of rolling a single die, find the event \"an odd number is rolled\". (a) {1, 3, 5} (b) {1, 2, 3} (c) {1, 5, 6} (d) {1, 4, 5} 5. In a random experiment of tossing a coin twice find the probability of getting exactly one head. (a) 1 4 (b) 2 3 (c) 4 3 (d) 1 2 6. The sample space for rolling a single die is given by S = {1, 2, 3, 4, 5, 6}. Define event A = {1, 3, 5} and consider the following events : i)B = {1, 2} ii)C = {2, 4} iii)D = {4} iv)Ac (a) i) iii) iv) are mutually exclusive with A (b) i) ii) are mutually exclusive with A (c) only iii) is mutually exclusive with A 1 (d) ii) iii) iv) are mutually exclusive with A Use the following information to answer the questions 7-12: Find the probabilities of the following using the information from the table given below: R S T M 0.09 0.25 0.19 N 0.31 0.16 0.00 7. P (R) = (a) 0.62 (b) 0.40 (c) 0.04 (d) 0.53 0.41 (b) 0.26 (c) 0.03 (d) 0.77 8. P (S) = (a) 9. P (R S) = (a) 0 (b) 0.1 (c) 1 (d) 0.2 10. P (Rc ) = (a) 0.60 (b) 0.42 (c) 0.33 (d) 0.15 0.72 (b) 0.46 (c) 0.81 (d) 0.32 11. P (R S) (a) 12. Determine whether or not the events N and S mutually exclusive (a) N and S are mutually exclusive (b) N and S are not mutually exclusive 2 Use the following information to answer the questions 13-16: The table relates the response to a fundraising appeal by a college to its alumni to the number of years since graduation An alumnus is selected at random. Adjoin the row and column totals to the table and use the expanded table to find the probability of each of the following events. 13. The alumnus responded (a) 0.72 (b) 0.460 (c) 0.181 (d) 0.0860 14. The alumnus did not respond (a) 0.1724 (b) 0.4196 (c) 0.8211 (d) 0.9140 15. The alumnus graduated at least 21 years ago. (a) 0.112 (b) 0.4500 (c) 0.3121 (d) 0.4411 16. The alumnus graduated at least 21 years ago and responded. (a) 0.033 (b) 0.1146 (c) 3 0.0081 (d) 0.0300 17. Identify the given random variable as either discrete or continuous: i) The number of passengers in a passenger vehicle on a highway at rush hour. ii) The air pressure of a tire on an automobile. iii) The weight of refuse on a truck arriving at a landfill. (a) i) ii) iii) are continuous (b) i) is discrete ii) and iii) are continuous (c) i) and ii) are discrete iii) continuous (d) None of the above 18. Identify the set of possible values for the the random variable X defined as follows: The number of heads in two tosses of a coin. (a) {0,1} {0} (b) (c) {0, 1, 2} {2} (d) Use the following information to answer question 19 - 21: The following table denotes the probability distribution for a discrete random variable X. X P (X) -2 0 1 2 9 0.1 0.3 0.2 0.3 0.1 19. Find P (X > 0). (a) 0.2 (b) 0.9 (c) 0.7 (d) 0.6 2.7 (d) 1.3 20. Find the expected value (mean) of X. (a) 2.0 (b) 1.5 (c) 21. The standard deviation of X is closest to (a) 17.50 (b) 2.77 (c) 4 7.65 (d) 4.18 22. Let X denote the number of boys in a randomly selected three-child family. Assuming that boys and girls are equally likely, construct the probability distribution of X. (a) x 1 2 3 4 x 0 (c) p 1/8 2/8 3/8 4/8 (b) x 0 1 2 2 3 4 p 0 2/6 3/6 4/6 3 x (d) p 1/8 3/8 3/8 1/8 1 2 3 4 p 1/16 2/16 3/16 4/16 Use the following information to answer questions 23 - 25: A discrete random variable X has the following probability distribution. x 13 18 20 24 27 p 0.22 0.25 0.20 0.17 0.16 Compute the following quantities. 23. P (X 18) = (a) 0.51 (b) 0.34 (c) 0.47 (b) 0.62 (c) 0.1 (d) 0.121 24. P (X > 20) = (a) 0.48 (d) 0.33 25. The variance 2 of X. (a) 21.822 (b) 14.511 (c) 22.282 (d) 19.863 26. A continuous random variable X has a uniform distribution on the interval [5, 12]. What is the density function of X? 7 (a) f (x) = , for 5 < x < 12 ; 0 elsewhere 9 1 (b) f (x) = , for 5 < x < 12 ; 0 elsewhere 7 1 (c) f (x) = , for 5 < x < 12 ; 0 elsewhere 9 5 (d) f (x) = , for 5 < x < 12 ; 0 elsewhere 12 5 27. Use the information from previous problem to find the probability of X assuming a value within the interval [8, 10]. (a) 7 5 2 7 (b) 3 4 (c) (d) 8 9 28. A continuous random variable X has a normal distribution with mean 169. The probability that X takes a value greater than 180 is 0.17. Find the probability that X takes a value less than 158. (a) 0.25 (b) 1.69 (c) 0.17 (d) 1.80 29. Let X be a continuous random variable with mean 12.25 and standard deviation 1.05. Find P (X = 14). (a) 0 (b) 2 (c) 3 (d) -4 30. A standard normal random variable has standard deviation equal to (a) 0 (b) 1 (c) -1 (d) 5 31. Find the probability P (0.62 < Z < 0.95) where Z is a standard normal random variable . (a) 0.5111 (b) 0.4134 (c) 0.5221 (d) 0.5613 32. For some z value, P (Z > z) = 0.125 then P (Z < z) = (a) 0.125 (b) 0.215 (c) 0.0125 (d) 0.25 33. P (Z < 1.96 orZ > 1.96) (a) 0.1970 (b) 0.0098 (c) 0.0500 (d) 0.0050 34. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the probability that a person selected at random will have an IQ score less than 112. (a) 0.4179 (b) 0.6901 (c) 6 0.7881 (d) 0.7213 35. Find the following probability of a general normal random variable X with mean 72 and standard deviation 22. P (X < 50 orX > 94) (a) 0.5331 (b) 0.2612 (c) 0.1121 (d) 0.3173 36. P (Z > z ) = 0.8200 (a) -0.31 (b) 0.92 (c) -0.92 (d) -0.28 37. X is a normally distributed random variable with mean 15 and standard deviation 0.25. Find the values xL and xR of X that are symmetrically located with respect to the mean of X and satisfy P (xL < X < xR ) = 0.80 (a) xL = 13.68; xR = 14.52 (c) xL = 14.68; xR = 15.32 (b) xL = 11.18; xR = 12.92 (d) xL = 10.83; xR = 18.65 7