Worksheet #3: Unions and Intersections Answer Key
Worksheet #3: Unions and Intersections Answer Key Name_____________________________________________ c 1. If P(X) = 0.23 and P(X and Y) = 0.12 and P(X or Y) = .34, find P(Y ). X 0.11 Y 0.12 0.11 P A B 2. Let 3 6 P Ac B 1 6 P A Bc 1 6 (a) Draw and label a Venn Diagram for events A and B. Make a Venn Diagram. (b) P Ac Bc = 1 – 1/6 – 3/6 – 1/6 = 1/6 (c) P Ac Bc = 1 – 3/6 = 3/6 (or 1/2) (d) Are A and B mutually exclusive (disjoint)? No, they have an intersection. (e) Are A and B independent? Is P(A)P(B) = P(A and B)? 4/9 is not equal to 3/6, so no. Therefore, they are not independent. 3. Suppose the probability that a consulting company will be awarded a certain contract is 0.25, the probability that it will be awarded a second contract is 0.21, and the probability that it will get both contracts is 0.13. What is the probability that the company will win at least one of the two contracts? Make a Venn Diagram. 1 – 0.67 = 0.33 4. The probability that a graduate student will receive a state grant is 1/3 while the probability that she will be awarded a federal grant is 1/2. If whether or not she receives one grant is not influenced by whether or not she receives the other, what is the probability of her receiving both grants? Independent events, so (1/3)(1/2) = 1/6 5. Suppose that among the 6000 students at a high school, 1500 are taking honors courses and 1800 prefer watching basketball to watching football. If taking honors courses and preferring basketball are independent, how many students are both taking honors courses and prefer basketball to football? Independent events, so (1500/6000)(1800/6000) = 3/40 or 0.075 Worksheet #3: Unions and Intersections Answer Key 6. Suppose that, for any given year, the probabilities that the stock market declines, that women’s hemlines are lower, and that both events occur are, respectively .4, .35, and .3. Are the two events independent? Is (0.4)(0.35) = 0.3? No. So the events are not independent. Use the table for questions 7 – 10. GPA Many Skipped Classes Few Skipped Classes Total < 2.0 80 175 255 2.0–3.0 25 450 475 > 3.0 5 265 270 Total 110 890 1000 7. What is the probability that a student has a GPA between 2.0 and 3.0? 475/1000 = 0.475 8. What is the probability that a student has a GPA under 2.0 and has skipped many classes? 80/1000 = 0.08 9. What is the probability that a student has a GPA under 2.0 or has skipped many classes? (255 + 110 – 80)/1000 = 0.285 10. Are “GPA between 2.0 and 3.0” and “Skipped Few Classes” independent? Show how you know. (0.475)(0.890) is not equal to 0.450, so not independent 11. Given the probabilities P(A) = .4 and P( A B ) = .6, what is the probability P(B) if A and B are mutually exclusive? If A and B are independent? If mutually exclusive: 0.6 – 0.4 = 0.2 If independent … 0.6 = 0.4 + P(B) – 0.4(P(B)) 0.2 = 0.6(P(B)) 1/3 = P(B) MULTIPLE CHOICE PRACTICE 12. Suppose the probability that you will receive an A in AP Statistics is .35, the probability that you will receive A’s in both AP Statistics and AP Biology is .19, and the probability that you will receive an A in AP Statistics but not in AP Biology is .17. Which of the following is a proper conclusion? 0.35 – 0.19 is not equal to 0.17, so the answer is D (a) The probability that you will receive an A in AP Biology is .36. (b) The probability that you didn’t take AP Biology is .01. (c) The probability that you will receive an A in AP Biology but not in AP Statistics is .18. (d) The given probabilities are impossible. (e) None of the above 13. If P(A) = .2 and P(B) = .1, what is P( A B ) if A and B are independent? 0.2 + 0.1 – (0.2)(0.1) = 0.28, so B (a) .02 (b) .28 (c) .30 (d) .32 (e) There is insufficient information to answer this question. Worksheet #3: Unions and Intersections Answer Key 14. Among a group of boys, 70% like chocolate ice cream, 40% like strawberry ice cream, and 30% like both. If a boy is randomly selected from the group, what is the probability that he likes either chocolate or strawberry ice cream, but not both? Use a Venn Diagram. 0.4 + 0.1 = 0.5, so D (a) 10% (b) 20% (c) 30% (d) 50% (e) 80% 15. At Walton High School, 5% of athletes play both football and some other contact sport, 30% play football, and 40% play other contact sports. If there are 200 athletes, how many play neither football nor any other contact sport? Use a Venn Diagram. (0.35)(200) = 70, so B (a) 20 (b) 70 (c) 80 (d) 100 (e) 130 16. If P(A) = 0.4, P(B) = 0.2, and P(A and B) = 0.08, which of the following is true? (0.4)(0.2) = 0.08, so they are independent. Independent events cannot be mutually exclusive (disjoint), so B. (a) Events A and B are independent and mutually exclusive. (b) Events A and B are independent but not mutually exclusive. (c) Events A and B are mutually exclusive but not independent. (d) Events A and B are neither independent nor mutually exclusive. (e) Events A and B are independent but whether A and B are mutually exclusive cannot be determined from the given information. 17. Let x = P(A), y = P(B), and z = P(A B). Which of the following facts would indicate that events A and B are dependent events? A I. xy < z II. z > 0 III. xy > 0 (a) I only (b) I and III (c) II and III (d) I, II, and III (e) None of the facts would indicate dependence. 18. Given two independent events, X and Y, such that P(Y) = 0.2 and P(X Y) = 0.4, what is the value of P(X)? Same process as #11. The answer is C. (a) 0.05 (b) 0.20 (c) 0.25 (d) 0.30 (e) Cannot be determined from the information given. 19. Given that 52% of the U.S. population are female and 15% are older than age 65, can we conclude that (.52)(.15) = 7.8% are women older than age 65? Women, on average, live longer than men, so these two events are not independent. The answer is D. (a) Yes, by the multiplication rule (b) Yes, by conditional probabilities (c) Yes, by the law of large numbers (d) No, because the events are not independent (e) No, because the events are not mutually exclusive Worksheet #3: Unions and Intersections Answer Key 20. If AB = S (sample space), P(A and Bc) = 0.25, and P(Ac) = 0.35, then P(B) = Make a Venn Diagram. The answer is D. (a) 0.35. (b) 0.4. (c) 0.65 (d) 0.75. (e) None of the above.
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