MATH 151, Section A01 Test 1 Version 1 February 14, 2013

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1 MATH 151, Section A01 Test 1 Version 1 February 14, 2013 Last Name, First Name: Student Number: Instructor: Dr. Jing Huang TO BE ANSWERED ON THE PAPER THIS TEST HAS 6 PAGES OF QUESTIONS, PLUS COVER. Instructions: Your NAME and STUDENT NUMBER must be recorded on your test paper. The only hand calculator permitted is the Sharp EL-510R. No other electronic devices are permitted. Questions 1 through 8 are multiple choice. For questions requiring numerical answers, the choices are listed in numerically increasing order. Choose the value that is nearest your unrounded answer and clearly circle your selection. If your answer is the same distance from the two nearest choices, choose the larger of the two choices. For verification purposes, show all calculations on your question paper. Unverified answers may be disallowed. Questions 9 and 10 are full-answer questions. For these questions, write out your solutions carefully and completely on the question paper. Marks will be deducted for incomplete or poorly presented solutions. Each multiple-choice question is worth 2 marks. Questions 9 and 10 are worth 4 marks each. Maximum score is 24 marks. Your total score: /24

2 1. On a particular morning, the university cafeteria served 425 people breakfast. At lunch time they served 600 people lunch and asked each of them whether they had also eaten breakfast there that morning. There were 280 people who said they had also eaten breakfast there that morning. How many people ate exactly one of breakfast or lunch at the cafeteria on this day? (A) 220 (B) 280 (C) 340 (D) 400 (E) 460 (F) 520 (G) 580 (H) 640 (I) 700 (J) 760 ANS = 465 E 2. Joe has 4 nonfiction and 7 fiction books on his desk that he has not yet read. He is going on a relaxing vacation and plans to pack 3 of these books to take with him. How many possible selections would include at least one fiction and one nonfiction book? (A) 10 (B) 25 (C) 50 (D) 75 (E) 125 (F) 250 (G) 500 (H) 1000 (I) 2000 (J) 3000 ANS = 126 E

3 3. Lennox has 5 important tasks that he needs to complete at work today. One of them must be done immediately so he decides to do that one first. How many possibilities are there for the order in which Lennox will do the 5 tasks? (A) 10 (B) 25 (C) 50 (D) 75 (E) 125 (F) 250 (G) 500 (H) 1000 (I) 2000 (J) 3000 ANS = 24 B 4. How many of the linear arrangements of the 10 letters in the word S C I E N T I F I C have all 3 Is adjacent as well as have the 2 Cs adjacent? An example of such an arrangement is S C C N F I I I E T. (A) 100 (B) 500 (C) 1, 000 (D) 5, 000 (E) 10, 000 (F) 50, 000 (G) 100, 000 (H) 200, 000 (I) 300, 000 (J) 500, 000 ANS = 5040 D

4 5. Four friends are each asked to think of a number between 1 and 10 (one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and write it down on a slip of paper along with their name. How many different outcomes are there for the numbers that the four people write down? (A) 1, 000 (B) 5, 000 (C) 10, 000 (D) 50, 000 (E) 100, 000 (F) 500, 000 (G) 1, 000, 000 (H) 2, 000, 000 (I) 3, 000, 000 (J) 4, 000, 000 ANS = 10,000 C 6. Suppose A and B are independent events of an experiment, Pr(A) = 0.28, and Pr(B) = Find the probability Pr(B A ) (note the complement A ). (A) 0.25 (B) 0.30 (C) 0.35 (D) 0.40 (E) 0.45 (F) 0.50 (G) 0.55 (H) 0.60 (I) 0.65 (J) 0.70 ANS = 0.65 I

5 7. The city s Air Quality Board monitors daily levels of pollen and ozone (since these are known to be contributors to allergy irriation for people). A warning is given when levels go above some threshold. Ozone levels are above the threshold on 30% of days, and pollen levels are above the threshold on 20% of days. On 6% of days, both ozone and pollen levels are above the threshold. Let E denote the event that ozone levels are above the threshold and F denote the event that pollen levels are above the threshold. Determine which of the following statements are correct: (i) E and F are independent and mutually exclusive. (ii) E and F are independent but not mutually exclusive. (iii) E and F are mutually exclusive but not independent. (A) None (B) Only (i) (C) Only (ii) (D) Only (iii) (E) (i) & (ii) (F) (i) & (iii) (G) (ii) & (iii) (H) All ANS = C 8. A random variable X has the following probability distribution: Calculate E(X). x P r(x = x) (A) 0 (B) 0.5 (C) 1.0 (D) 1.5 (E) 2.0 (F) 2.5 (G) 3.0 (H) 4.0 (I) 6.0 (J) 8.0 ANS = 2.1 E

6 The following questions require full answers. Answer each question in the space provided. 9. A game is played as follows: a player is dealt 2 cards from a standard deck of 52 cards. If the dealt contains no Hearts then the player is given a single coin to flip; if the dealt contains exactly one Heart then the player is given two coins to flip; if both cards are Hearts then the player is given three coins to flip. The player wins $10 for each coin that lands Heads. If you know that a person who played this game did not win any money, what is the probability that the person was not dealt any Hearts? Denote H 0 no heart, H 1 one heart, H 2 two hearts, and W win no money. Then Pr(H 0 ) = 19/34, Pr(H 1 ) = 13/34, Pr(H 2 ) = 1/17, Pr(W H 0 ) = 1/2, Pr(W H 1 ) = 1/4, and Pr(W H 2 ) = 1/8. By Bayes theorem, Pr(H 0 W) = = Pr(H 0 )Pr(W H 0 ) Pr(H 0 )Pr(W H 0 )+Pr(H 1 )Pr(W H 1 )+Pr(H 2 )Pr(W H 2 ) (19/34)(1/2) (19/34)(1/2)+(13/34)(1/3)+(1/17)(1/8) = (19)(2) (19)(2)+13+1 = 38/

7 10. A test consist of 8 multiple-choice questions, each of which has 10 answer options. A particular student did not study and decides to simply guess on each of the questions, each time choosing one of the 10 options at random. (a) Find the probability that the student will guess correctly on at least 2 of the questions. Let X be the random variable of the number of questions guessed correctly. Then Pr(X 2) = 1 Pr(X = 0) Pr(X = 1) = 1 ( ) 8 0 (1/10) 0 (9/10) 8 ( ) 8 (1/10) 1 (9/10) (b) What is the expected value of the student s test score? E(X) = 8(1/10) =.8 - END -