Math 147 Elementary Probability/Statistics I Additional Exercises on Chapter 4: Probability Student Name: Find the indicated probability. 1) If you flip a coin three times, the possible outcomes are HHH HHT HTH HTT THH THT TTH TTT. What is the probability of getting at least one head? 2) If you flip a coin three times, the possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. What is the probability that the first two tosses come up the same? 3) If two balanced die are rolled, the possible outcomes can be represented as follows. (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) (1, 4) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4) (1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5) (1, 6) (2, 6) (3, 6) (4, 6) (5, 6) (6, 6) Determine the probability that the sum of the dice is 2 or 10. 4) A committee of three people is to be formed. The three people will be selected from a list of five possible committee members. A simple random sample of three people is taken, without replacement, from the group of five people. Using the letters A, B, C, D, E to represent the five people, list the possible samples of size three and use your list to determine the probability that B is included in the sample. (Hint: There are 10 possible samples.) 5) A bag contains four chips of which one is red, one is blue, one is green, and one is yellow. A chip is selected at random from the bag and then replaced in the bag. A second chip is then selected at random. Make a list of the possible outcomes (for example RB represents the outcome red chip followed by blue chip) and use your list to determine the probability that the two chips selected are the same color. (Hint: There are 16 possible outcomes.) 6) A 12-sided die is rolled. What is the probability of rolling a number less than 11? 7) In a poll, respondents were asked whether they had ever been in a car accident. 233 respondents indicated that they had been in a car accident and 311 respondents said that they had not been in a car accident. If one of these respondents is randomly selected, what is the probability of getting someone who has been in a car accident? 1
8) A survey resulted in the sample data in the given table. If one of the survey respondents is randomly selected, find the probability of getting someone who lives in a flat. Type of accommodation Number House 628 Flat 208 Apartment 262 Other 495 Find the indicated probability by using the special addition rule. 9) The age distribution of students at a community college is given below. Age (years) Number of students (f) Under 21 412 21-25 407 26-30 206 31-35 52 Over 35 20 1097 A student from the community college is selected at random. Find the probability that the student is between 26 and 35 inclusive. Round approximations to three decimal places. 10) A relative frequency distribution is given below for the size of families in one U.S. city. Size Relative frequency 2 0.439 3 0.230 4 0.209 5 0.070 6 0.036 7+ 0.016 A family is selected at random. Find the probability that the size of the family is less than 5. Round approximations to three decimal places. 11) A percentage distribution is given below for the size of families in one U.S. city. Size Percentage 2 44.3 3 24.2 4 15.3 5 12.0 6 2.7 7+ 1.5 A family is selected at random. Find the probability that the size of the family is at most 3. Round approximations to three decimal places. 2
12) The distribution of B.A. degrees conferred by a local college is listed below, by major. Major Frequency English 2073 Mathematics 2164 Chemistry 318 Physics 856 Liberal Arts 1358 Business 1676 Engineering 868 9313 What is the probability that a randomly selected degree is in English or Mathematics? 13) Two 6-sided dice are rolled. What is the probability that the sum of the numbers on the dice is 6 or 10? Find the indicated probability by using the general addition rule. 14) When two balanced dice are rolled, there are 36 possible outcomes. Find the probability that either doubles are rolled or the sum of the dice is 6. 15) In one city, 47.2% of adults are female, 10.2% of adults are left-handed, and 4.7% are left-handed females. For an adult selected at random from the city, let F = event the person is female L = event the person is left-handed. Find P(F or L). Round approximations to three decimal places. 16) Let A and B be events such that P(A) = 1 6, P(A or B) = 1 5, and P(A and B) = 1. Determine P(B). 11 17) A spinner has regions numbered 1 through 15. What is the probability that the spinner will stop on an even number or a multiple of 3? 18) Of the 51 people who answered "yes" to a question, 7 were male. Of the 100 people who answered "no" to the question, 13 were male. If one person is selected at random from the group, what is the probability that the person answered "yes" or was male? 3
Find the indicated probability by using the complementation rule. 19) The age distribution of students at a community college is given below. Age (years) Number of students (f) Under 21 411 21-24 418 25-28 268 29-32 141 33-36 104 37-40 59 Over 40 90 1491 A student from the community college is selected at random. Find the probability that the student is 21 years or over. Give your answer as a decimal rounded to three decimal places. 20) A relative frequency distribution is given below for the size of families in one U.S. city. Size Relative frequency 2 0.410 3 0.235 4 0.198 5 0.102 6 0.036 7+ 0.019 A family is selected at random. Find the probability that the size of the family is at most 6. Round approximations to three decimal places. 21) A percentage distribution is given below for the size of families in one U.S. city. Size Percentage 2 44.2 3 21.5 4 21.0 5 7.8 6 4.0 7+ 1.5 A family is selected at random. Find the probability that the size of the family is 4 or more. Round results to three decimal places. 22) Based on meteorological records, the probability that it will snow in a certain town on January 1st is 0.307. Find the probability that in a given year it will not snow on January 1st in that town. Solve the problem. 23) There are 30 chocolates in a box, all identically shaped. There 11 are filled with nuts, 10 with caramel, and 9 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting 2 solid chocolates in a row. 4
24) An ice chest contains 9 cans of apple juice, 8 cans of grape juice, 7 cans of orange juice, and 5 cans of pineapple juice. Suppose that you reach into the container and randomly select three cans in succession. Find the probability of selecting no grape juice. 25) Numbered disks are placed in a box and one disk is selected at random. If there are 8 red disks numbered 1 through 8, and 2 yellow disks numbered 9 through 10, find the probability of selecting a red disk, given that an odd-numbered disk is selected. 26) Four students drive to school in the same car. The students claim they were late to school and missed a test because of a flat tire. On the makeup test, the instructor asks the students to identify the tire that went flat; front driver's side, front passenger's side, rear driver's side, or rear passenger's side. If the students didn't really have a flat tire and each randomly selects a tire, what is the probability that all four students select the same tire? 27) A fast-food restaurant chain with 700 outlets in the United States describes the geographic location of its restaurants with the accompanying table of percentages. A restaurant is to be chosen at random from the 700 to test market a new style of chicken. Region NE SE SW NW <10,000 3% 6% 3% 0% Population of City 10,000-100,000 15% 1% 12% 5% >100,000 20% 4% 5% 26% Given that the restaurant is located in the eastern United States, what is the probability it is located in a city with a population of at least 10,000? 28) The breakdown of workers in a particular state according to their political affiliation and type of job held is shown here. Suppose a worker is selected at random within the state and the worker's political affiliation and type of job are noted. Political Affiliation Republican Democrat Independent White collar 20% 10% 18% Type of job Blue Collar 15% 19% 18% Given the worker is a Democrat, what is the probability that the worker is in a white collar job. 29) A single die is rolled twice. Find the probability of getting a 2 the first time and a 2 the second time. 30) If you toss a fair coin 11 times, what is the probability of getting all heads? 31) A human gene carries a certain disease from the mother to the child with a probability rate of 54%. That is, there is a 54% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has four children. Assume that the infections of the four children are independent of one another. Find the probability that all four of the children get the disease from their mother. 5
32) Suppose a basketball player is an excellent free throw shooter and makes 90% of his free throws (i.e., he has a 90% chance of making a single free throw). Assume that free throw shots are independent of one another. Suppose this player gets to shoot four free throws. Find the probability that he misses all four consecutive free throws. 33) A human gene carries a certain disease from the mother to the child with a probability rate of 51%. That is, there is a 51% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has three children. Assume that the infections of the three children are independent of one another. Find the probability that at least one of the children get the disease from their mother. 34) Investing is a game of chance. Suppose there is a 36% chance that a risky stock investment will end up in a total loss of your investment. Because the rewards are so high, you decide to invest in three independent risky stocks. Find the probability that at least one of your three investments becomes a total loss. 35) Numbered disks are placed in a box and one disk is selected at random. If there are 6 red disks numbered 1 through 6, and 4 yellow disks numbered 7 through 10, find the probability of selecting a yellow disk, given that the number selected is less than or equal to 3 or greater than or equal to 8. 36) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a company's success has been proven time and again to be customer service. A study was conducted to study the customer satisfaction levels for one overnight shipping business. In addition to the customer's satisfaction level, the customers were asked how often they used overnight shipping. The results are shown below in the following table: Frequency of Use High Satisfaction level Medium Low TOTAL < 2 per month 250 140 10 400 2-5 per month 140 55 5 200 > 5 per month 70 25 5 100 TOTAL 460 220 20 700 A customer is chosen at random. Given that the customer uses the company less than two times per month, what is the probability that they expressed high satisfaction with the company? 37) A study was recently done that emphasized the problem we all face with drinking and driving. Four hundred accidents that occurred on a Saturday night were analyzed. Two items noted were the number of vehicles involved and whether alcohol played a role in the accident. The numbers are shown below: Number of Vehicles Involved Did Alcohol Play a Role? 1 2 3 or more Totals Yes 54 91 25 170 No 20 172 38 230 Totals 74 263 63 400 Given that an accident involved multiple vehicles, what what is the probability that it involved alcohol? 6
38) The manager of a used car lot took inventory of the automobiles on his lot and constructed the following table based on the age of his car and its make (foreign or domestic): Age of Car (in years) Make 0-2 3-5 6-10 over 10 Total Foreign 41 21 13 25 100 Domestic 36 29 10 25 100 Total 77 50 23 50 200 A car was randomly selected from the lot. Given that the car selected was a foreign car, what is the probability that it was older than 2 years old? 39) A fast-food restaurant chain with 700 outlets in the United States describes the geographic location of its restaurants with the accompanying table of percentages. A restaurant is to be chosen at random from the 700 to test market a new style of chicken. Region NE SE SW NW <10,000 3% 6% 3% 0% Population of City 10,000-100,000 15% 4% 12% 5% >100,000 20% 4% 10% 18% Given that the restaurant is located in the eastern United States, what is the probability it is located in a city with a population of at least 10,000? 40) The breakdown of workers in a particular state according to their political affiliation and type of job held is shown here. Suppose a worker is selected at random within the state and the worker's political affiliation and type of job are noted. Political Affiliation Republican Democrat Independent White collar 16% 7% 17% Type of job Blue Collar 19% 8% 33% Given the worker is a Democrat, what is the probability that the worker is in a white collar job. 41) Assume that P(A) = 0.7 and P(B) = 0.2. If A and B are independent, find P(A and B). 42) If P(A) = 0.72, P(B) = 0.11, and A and B are independent, find P(A B). 43) Given that events C and D are independent, P(C) = 0.3, and P(D) = 0.6, are C and D mutually exclusive? 44) Given events C and D with probabilities P(C) = 0.3, P(D) = 0.2, and P(C and D) = 0.1, are C and D independent? 7
Answer Key Testname: MATH 147 ADDITIONAL EXERCISES ON CHAPTER 4 1) 7 8 2) 1 2 3) 1 9 4) 3 5 5) 1 4 6) 5 6 7) 0.428 8) 0.131 9) 0.235 10) 0.878 11) 0.685 12) 0.455 13) 2 9 14) 5 18 15) 0.527 41 16) 330 17) 2 3 18) 0.424 19) 0.724 20) 0.981 21) 0.343 22) 0.693 12 23) 145 24) 95 261 25) 4 5 26) 1 64 27) 0.816 28) 0.345 1 29) 36 30) 1 2048 8
Answer Key Testname: MATH 147 ADDITIONAL EXERCISES ON CHAPTER 4 31) 0.085 32) 0.0001 33) 0.882 34) 0.7379 35) 1 2 36) 5 8 37) 38) 58 163 59 100 39) 0.827 40) 0.467 41) 0.14 42).072 43) no 44) no 9