Math 166 WIR, Fall 2011,cBenjamin AurispaMath 166 Exam 2 ReviewSections 2.1-2.4 & 3.1-3.4Note: Although this review highlights all sections covered on the exam, it is slightly more heavily weightedon the new material this week: Sections 3.2-3.4. For additional practice problems on previous material,please take a look at Week in Reviews 4 and 5.1. Calculate the following probabilities. (Z is the standard normal random variable.)(a) P (−0.33 < Z < 0.47)(b) P (Z ≤ 0.93)2. Find the value of a given the following probabilities.(a) P (Z < a) = 0.4542(b) P (Z ≥ a) = 0.9778(c) P (−a ≤ Z < a) = 0.52541Math 166 WIR, Fall 2011,cBenjamin Aurispa3. Suppose that the weights of students at a university are normally distributed with a mean of 165 lband a standard deviation 24 lb. What is the probability that a student selected at random(a) weighs between 150 and 200 pounds inclusive?(b) weighs more than 215 pounds?(c) What weight corresponds to the 70th percentile?4. The government of a certain country wants to create a system for tax purposes where families areclassified as “elite,” “upper class,” “middle class,” or “lower class” based on the total family income.Suppose family incomes in this country are normally distributed with a mean of $30,000 and a standarddeviation of $9,000. If this government knows that they want 15% elite, 25% upper class, 40% middleclass, and 20% lower class, what would be the range of income classified as “middle class?”2Math 166 WIR, Fall 2011,cBenjamin Aurispa5. There are 130 boxes of Cheerios in a grocery store. The number of Cheerios was counted in each box.The data is below.Number of Cheerios 510 480 467 434 521 535Number of Boxes 23 9 17 30 40 11Find the mean, median, mode, standard deviation, and variance for the number of Cheerios in a box.6. Consider the following probability distribution.X 1 2 3 4 5Probability 0.2 0.12 0.3 0.22(a) Fill in the missing probability and draw a probability histogram for X.(b) Find E(X), σ, V ar(X), and the mode.(c) Find P (2 ≤ X < 5).3Math 166 WIR, Fall 2011,cBenjamin Aurispa7. A game costs $3 to play. The game involves drawing two cards at random from a deck of cards. If apair of aces is drawn, you win $38. If any other pair is drawn, then you win $13. If two cards of thesame suit are drawn, you win $8. Otherwise, you win nothing. Let X be the net winnings for a personwho plays this game.(a) Find the probability distribution of X.(b) What are the expected net winnings (rounded to the nearest cent)? Is this game fair?8. A car insurance policy covers damages from a car accident. If you get in a major wreck, the insurancecompany pays out $3000. If you get in a minor wreck, the insurance company pays out $1000.(a) Suppose your monthly payment is $110. The probability that you get in a major wreck in agiven month is 0.01 and the probability you get in a minor wreck is 0.07. What is the insurancecompany’s expected gain?4Math 166 WIR, Fall 2011,cBenjamin Aurispa(b) Suppose that you provide an extra risk to the insurance company because the probability thatyou get in a minor wreck jumps to 0.15. (The probability you get in a major wreck stays thesame.) What can you expect your minimum monthly payment, $a, to be?9. Suppose that in a certain country, the probability a person is over 6-ft tall is 0.58. If a group of 40people is randomly selected from this country, what is the probability that(a) Exactly half of them are over 6-ft tall?(b) At least 24 of them are over 6-ft tall?(c) More than 15 but fewer than 21 are over 6-ft tall?(d) How many people in this group can you expect to be over 6-ft tall?(e) What is the standard deviation for the number of people in this group over 6-ft tall?5Math 166 WIR, Fall 2011,cBenjamin Aurispa10. A toy chest contains 9 Micro Machines, 7 Lego blocks, and 3 GI Joes.(a) Suppose that 6 toys are selected at random from the box.i. How many samples would contain exactly 3 Lego blocks and exactly 2 GI Joes?ii. What is the probability that exactly 3 Micro Machines or exactly 2 Lego blocks are selected?iii. What is the probability that at least 1 Micro Machine is selected?iv. Find the expected number of GI Joes in a sample.6Math 166 WIR, Fall 2011,cBenjamin Aurispa(b) If all 19 toys are to be arranged in a row, in how many ways can this be done? (Assume each toyis distinguishable.)(c) In how many ways can 5 of the 19 be arranged in a row?(d) In how many ways can these toys be arranged in a row if each type of toy is grouped together?(e) In how many ways can 9 of these toys be arranged in a row if a Micro Machine must be on eachend and the 3 GI Joes must be in the middle?(f) If toys of each type are identical, in how many distinguishable ways can all the toys be arranged?(g) If the Micro Machines and Legos are identical, but the GI Joes are not, in how many distinguish-able ways can all the toys be arranged?7Math 166 WIR, Fall 2011,cBenjamin Aurispa11. A 7-character code consists of 2 letters, followed by 4 digits, followed by another letter.(a) How many codes are possible?(b) How many codes are possible if the first letter must be a consonant, the last letter must be avowel, and no letter or digit can be repeated?(c) How many codes are possible if the code cannot start with M, the first two digits cannot be 0,and no letter or digit can be repeated?12. There are 10 freshmen, 12 sophomores, 7 juniors, and 4 seniors in a high school class. The StudentCouncil consists of a President, Vice President, Secretary, a 4-person social subcommittee, and a 3-person prom-planning subcommittee. A student cannot hold more than one position or be on morethan one committee. How many Student Councils can be formed if(a) there are no restrictions.(b) the president and vice president must be seniors, and the prom-planning committee must consistof all juniors.8Math 166 WIR, Fall 2011,cBenjamin Aurispa13. Determine the possible values of X for the following random variables and classify the random variableas Finite Discrete, Infinite Discrete, or Continuous.(a) An experiment consists of rolling a fair 6-sided die 4 times. Let X be the sum of the numbersrolled.(b) An experiment consists of drawing cards one at a time without replacement from a standard deckuntil all four Aces have been drawn. Let X be the number of draws needed.(c) A bag of marbles contains 5 reds and 6 blues. An experiment consists of pulling marbles one ata time with replacement until a red marble is