1 Probability: General Rules, Independence, and Conditional Probability 1. A high school science teacher has 78 students. Of those students, 35 are in the band and 32 are on a sports team. There are 16 students who are not in the band or on a sports team. One student from the 78 students will be selected at random. Let event B represent the event of selecting a student in the band, and let event S represent the event of selecting a student on a sports team. Are B and S mutually exclusive events? Steam. Ar 32 A No, because P(BnS) = . 5 78. 16 B. No, because P(BnS) = 48 78. B S C. No, because P(B ∩S) = 62 78 D. Yes, because P(BnS) = 5 78 E. Yes, because P(BnS) = %. 62 78. PCAUB) = 78-16=62 - 78 78 PLAUB) = P(A)+PCB) - PANB). 62 To = 35 32 18+ 78 - X 78 Page 2 Team Home Away Total 2. Purchased food 120 40 160 Did not purchase food 60 30 90 Total 180 70 250 The table shows data that were collected from people who attended a certain high school basketball game. A person who attended the game will be selected at ran- dom. Which of the following correctly interprets mutually exclusive events repre- sented by the table? A. Rooting for the home team and rooting for the away team B. Rooting for the home team and purchasing food during the game C. Rooting for the away team and purchasing food during the game D. Rooting for the home team and not purchasing food during the game E. Not rooting for the home team and not purchasing food during the game Page 2 3. A middle school chess club has 5 members: Adam, Bradley, Carol, Dave, and Ella. Two students from the club will be selected at random to participate in the county chess tournament. What is the probability that Adam and Ella will be selected? A. 1 20 B. 1 10 A B C DE C. ハハハハ D. B CDE ACDE ABDE ABCE ABLD Ε. Page 4 2 二 4. Events D and E are independent, with P(D) = 0.6 and P(D and E) = 0.18. Which of the following is true? A. P(E) = 0.12 B. P(E) = 0.4 C. P(D or E) = 0.28 D. P(D or E) = 0.72 E. P(D or E) = 0.9 Page 5 5. One student from a high school will be selected at random. Let A be the event that the selected student is a student athlete, and let B be the event that the selected student drives to school. If P(A ∩ B) = 0.08 and P(B | A) = 0.25, what is the probability that the selected student will be a student athlete? A. 0.02 B. 0.17 C. 0.32 D. 0.33 Ε. 3.13 PLANB) = PLA). PCBIA) 0.08 0.25 ↑ ? 0.08 PLA) = 0-25:0.32 Page 6 6. Each of the faces of a fair six-sided number cube is numbered with one of the numbers 1 through 6, with a different number appearing on each face. Two such number cubes will be tossed, and the sum of the numbers appearing on the faces that land up will be recorded. What is the probability that the sum will be 4, given that the sum is less than or equal to 6? A. 2/36 Β. 3/36 C. 3/15 D. 2/9 Ε. 4/6 X.Y RV. P(X+Y=4 / X+Y26) 11-5 x5 21-4 x4 31-3 x2 4 1-2 23 51 136 こ Pogo 7 P(x+y=4}{X+Y=6}) 3 P(X+Y=6) 36 →PCX+Y=4) PCX+Y50) 36 7. As a promotion, the first 50 customers who entered a certain store at a mall were asked to choose from one of two discounts. The first discount choice was 20% off all purchases made that day. The second discount choice was 10% off all purchases for the week. Of those who received the discounts, 28 chose the first discount and 22 chose the second discount. One customer will be selected at random from those who received a discount. Let F represent the event that the selected person chose the first discount, and let S represent the event that the selected person chose the second discount. Are F and S mutually exclusive events? A. Yes, because P(F∩S) = 0. B. Yes, because P(F∩S) = 0.12. C. Yes, because P(F∩S) = 1. D. No, because P(F∩S) = 0. E. No, because P(F ∩ S) = 1. Page 8 8. For flights from a particular airport in January, there is a 30 percent chance of a flight being delayed because of icy weather. If a flight is delayed because of icy weather, there is a 10 percent chance the flight will also be delayed because of a mechanical problem. If a flight is not delayed because of icy weather, there is a 5 percent chance that it will be delayed because of a mechanical problem. If one flight is selected at random from the airport in January, what is the probability that the flight selected will have at least one of the two types of delays? A. 0.065 Β. 0.335 C. 0.350 PLIUM) D. 0.450 PLI)=a3 Ε. 0.665 азт Di delay Mal изход MC PCM11)=0.1 0.79C005M Txos PLM 174)=0.05 Page 0 9. Ali surveyed 200 students at a school and recorded the eye color and the gender of each student. Of the 80 male students who were surveyed, 60 had brown eyes. If eye color and gender are independent, how many female students surveyed would be expected to have brown eyes? A. 5 B. 20 C. 30 D. 90 Ε. 100 Page 10 10. The SC Electric Company has bid on two electrical wiring jobs. The owner of the company believes that • the probability of being awarded the first job (event A) is 0.75; • the probability of being awarded the second job (event B) is 0.5; and • the probability of being awarded both jobs (event (A and B)) is 0.375. If the owner's beliefs are correct, which of the following statements must be true concerning event A and event B? A. Event A and event B are mutually exclusive and are independent. B. Event A and event B are mutually exclusive and are not independent. C. Event A and event B are not mutually exclusive and are independent. D. Event A and event B are not mutually exclusive and are not independent. E. Event A and event B are not mutually exclusive, and independence cannot be determined. Poco 11 11. For which of the following probability assignments are events A and B indepen- dent? A. P(A ∩ B) = 0.3. P(A ∩ B) = 0.12, and P(A∩ B) = 0.4. B. P(A∩ B) = 0.3, P(A∩B) = 0.3, and P(A∩B) = 0.3. C. P(A∩ Bº) = 0.1, P(A∩ B) =(0.1. and P(A∩ B) = 0.4. D. P(A ∩ Bº) = 0.3, P(A ∩ B) = 0.0, and P(A∩ B) = 0.2. E. P(A ∩ B) = 0.5, P(A ∩ B) = 0.1, and P(A∩ B) = 0.4. PCA∩B)=P(A).P(13). Pago 12 12. Ms. Tucker travels through two intersections with traffic lights as she drives to the market. The traffic lights operate independently. The probability that both lights will be red when she reaches them is 0.22. The probability that the first light will be red and the second light will not be red is 0.33. What is the probability that the second light will be red when she reaches it? A0.40 Β. 0.45 C. 0.50 D. 0.55 Ε. 0.60 PLS) =? =0.22. P(FAS)=222) P(FMS)=0.33 055 0.22=P(FAS)=PLF) XP(S). 0.22 PCS) = 0.5 = 0.4 • ** ? Pago 12 KA 13. In a certain school, 17 percent of the students are enrolled in a psychology course, 28 percent are enrolled in a foreign language course, and 32 percent are enrolled in either a psychology course or a foreign language course or both. What is the probability that a student chosen at random from this school will be enrolled in both a foreign language course and a psychology course? A. 0.45 B. 0.32 C. 0.20 D. 0.13 Ε. 0.05 917 P(44) = 0,170 VP(AUB)=432. P(B)=0,28 J VPCA1B)=? 0.28 P(AMB)=P(A). PCB) A&B independent - 0.32. P(A)+PCB) - PC/UB) = P(AMB). P(AUB)=P(A)+PCB) - PANB). Pogo 14 14. 14/ The The probability that a new microwave oven will stop working in less than 2 years is 0.05. The probability that a new microwave oven is damaged during delivery and stops working in less than 2 years is 0.04. The probability that a new microwave oven is damaged during delivery is 0.10. Given that a new microwave oven is damaged during delivery, what is the probability that it stops working in less than 2 years? A. 0.05 B. 0.06 C. 0.10 D. 0.40 P(A)= 0.05 PCAIB)=? 0.04 PCANB)=0.04. PCA1B7= PLANB) Ε. 0.50 P(B)=0.1. Pogo 15 PCB) 이 15. A student is applying to two different agencies for scholarships. Based on the stu- dent's academic record, the probability that the student will be awarded a schol- arship from Agency A is 0.55 and the probability that the student will be awarded a scholarship from Agency B is 0.40. Furthermore, if the student is awarded a scholarship from Agency A, the probability that the student will be awarded a scholarship from Agency B is 0.60. What is the probability that the student will be awarded at least one of the two scholarships? A. 0.60 B. 0.62 C. 0.71 D. 0.73 Ε. 0.95 2 Discrete Random Variables Page 16 1. The following table shows the probability distribution for the number of books a student typically buys at the annual book fair held at an elementary school. Number of Books 0 1 2 3 4 5 f 6 7 Probability 0.35 0.20 0.15 0.10 0.07 0.08 0.04 0.01 Let the random variable B represent the number of books a student buys at the next book fair. What is the expected value of B? A. 0 B. 1.00 C. 1.79 D. 3.50 Ε. 28 Page 17 2. At a large regional collegiate women's swim meet, an official records the time it takes each swimmer to swim 100 meters for all swimmers who compete in only one stroke category. The following table shows the mean times and corresponding standard deviations for the collegiate women at the swim meet for each of the four stroke categories. Stroke Category Mean 100 meter Time Standard Deviation Backstroke 55.6 seconds 0.70 seconds Breaststroke 63.3 seconds 0.92 seconds Butterfly 54.4 seconds 0.94 seconds Freestyle 50.2 seconds 0.76 seconds For each of the 4 stroke categories, consider a random variable representing the time of a randomly selected swimmer in that category. What is the standard deviation of the sum of the 4 random variables? A. 0.83 seconds B. 1.67 seconds C. 2.80 seconds D. 3.32 seconds E. 3.76 seconds Page 18 3. A player pays $15 to play a game in which a chip is randomly selected from a bag of chips. The bag contains 10 red chips, 4 blue chips, and 6 yellow chips. The player wins $5 if a red chip is selected, $10 if a blue chip is selected, and $20 if a yellow chip is selected. Let the random variable X represent the amount won from the selection of the chip, and let the random variable W represent the total amount won, where W = X – 15. What is the mean of W? A. $10.50 Β. $4.50 C. -$4.50 D. -$6.50 Ε. -$10.50 Page 10 4. A city department of transportation studied traffic congestion on a certain highway. To encourage carpooling, the department will recommend a carpool lane if the average number of people in passenger cars on the highway is less than 2. The probability distribution of the number of people in passenger cars on the highway is shown in the table. Number of people 1 2 3 4 5 Probability 0.56 0.28 0.08 0.06 0.02 Based on the probability distribution, what is the mean number of people in passenger cars on the highway? A. 0.28 B. 0.56 C. 1.7 D. 2 E. 3 Page 20 5. Every Thursday, Matt and Dave's Video Venture has