Math-138 Fall 2024 Unit 2 Review PDF

Math-138 Fall 2024 Unit 2 Review 1. Suppose you flip a coin 3 times. a. What is the sample space? b. What is the probability of observing exactly 2 heads? c. What is the probability of observing at least 2 heads? d. What...

Math-138 Fall 2024 Unit 2 Review 1. Suppose you flip a coin 3 times. a. What is the sample space? b. What is the probability of observing exactly 2 heads? c. What is the probability of observing at least 2 heads? d. What is the probability of observing more than 2 heads? e. What is the probability of observing at least 1 head? f. What is the probability observing exactly 1 head or exactly 3 tails? 2. On awards day, Jasmine has an 85% chance of winning the award in English and a 4 out of 5 chance of winning an award for athletics. What is the probability that she wins both awards? 3. The probability that Samantha will be accepted by the college of her choice and obtain a scholarship is 0.35. If the probability that the college accepts her is 0.65, find the probability that she will obtain a scholarship given that the college accepts her. 4. Suppose a jar contains 40 red marbles, 40 blue marbles and 20 green marbles. Find the following probabilities: a. If two marbles are chosen with replacement, what is the probability that both are green? b. If three marbles are chosen without replacement, what is the probability that none are green? c. If two marbles are chosen without replacement, what is the probability that the first marble is red and the second marble is blue? 1  Math-138 Fall 2024 d. If two marbles are chosen without replacement, given the first marble is red, what is the probability that the second marble is red? e. If two marbles are chosen without replacement, given the first marble is green, what is the probability that the second marble is blue? f. If five marbles are chosen without replacement, what is the probability that the 5th marble is the only marble that is green? 5. The table below shows whether students in an introductory statistics class like dogs and/or cats. (De Veaux et al., 2012) Like Dogs Yes No Total Like Yes 194 21 215 Cats No 110 10 120 Total 304 31 335 a. What percent of students like dogs? b. What percent of students like dogs and cats? c. What percent of students like dogs but not cats? d. What percent of students who like dogs also like cats? e. What percent of students who like cats do not like dogs? f. What percent of students like dogs or cats? g. Do “liking dogs” and “liking cats” appear to be independent? Give statistical evidence to support your conclusion. 2  Math-138 Fall 2024 6. They are 20 pencils in a desk drawer. 13 of them have erasers and 11 of them are mechanical. There are 4 mechanical pencils that have erasers. Using a Venn Diagram, find the following probabilities: a. Choosing a pencil at random from the drawer that is mechanical? b. Choosing a pencil at random from the drawer that is mechanical or has an eraser? c. Choosing a pencil at random from the drawer that is neither mechanical nor has an eraser? d. Choosing a pencil at random from the drawer that is mechanical, given that it has an eraser? e. Choosing a pencil at random from the drawer that has an eraser, given that it is mechanical? 7. Suppose that 70% of job postings on a career website require applicants to have a college degree, 50% of job postings require that an applicant have previous experience in the field, and that 35% of job postings require both, that applicants have a college degree and experience in the field. Draw a Venn Diagram and use your diagram to answer the questions below. Find: a. The probability that a random job posting will require applicants to have a degree or experience in the field. b. The probability that a random job posting will require neither that applicants have a degree nor that they have experience in the field. 3  Math-138 Fall 2024 c. The probability that a random job posting will require applicants to have a college degree but not experience in the field. d. The probability that a random job posting will require applicants to have a degree given that the posting requires applicants to have previous experience in the field. e. Are the events that a random job posting will require a college degree and will require previous experience independent? Show this using probability formulas. 8. In July 2008, a science journal published a report on the reliability of Alzheimer testing. Results of a large study suggested that among people with Alzheimer’s, 59% of the tests conducted were (correctly) positive. For people who do not have Alzheimer’s, 68% of the tests were (correctly) negative. A clinic serving an at-risk population offers free Alzheimer’s testing, believing that 15% of the patients may actually have Alzheimer’s disease. Draw a Tree Diagram and use it to answer the following questions. Find the following probabilities: a. P(a patient tested negative for Alzheimer’s and that patient does not have Alzheimer’s) b. P(a patient tested negative for Azheimer’s and that patient truly has Alzheimer’s) c. P(a patient test positive for Alzheimer’s and that patient does not have Alzheimer’s) d. P(a patient will test negative for Alzeheimer’s) 4  Math-138 Fall 2024 e. Given that a person tested negative for Alzheimer’s, find the P(that person is truly Alzheimer’s free). 9. In a game, a player can place a $7 bet on red and have a 0.4 probability of winning. If red turns up, the player wins $25; otherwise, the player loses. Create a valid probability distribution table and calculate the player’s expected gain or loss. 10. At a local university, 54.3% of incoming freshman have computers. If 3 students are randomly selected find the probability that: a. None have computers. b. At least one has a computer. c. All have computers. 11. It has been reported that 5% of Americans are afraid of being alone in a house at night. We are going to randomly select 20 Americans for this exercise. a. Find the probability that exactly 5 people in the sample we choose are afraid of being alone in a house at night. b. Find the probability that at most 4 people are afraid. c. Find the probability that at least 3 people are afraid. d. Find the probability that no more than 2 people are afraid. e. Find the probability that less than 5 people are afraid. f. Find the probability that no less than 5 people are afraid. 5  Math-138 Fall 2024 g. Find the probability that between 3 and 5 people are afraid (inclusive). h. Find the mean and the standard deviation of this distribution. 12. The costs for standard veterinary services at a local animal hospital follow a Normal model with a mean of $90 and a standard deviation of $30. (De Veaux et al., 2012) Diagrams are recommended. a. What percentage of standard veterinary bills will be greater than $175? b. What percentage of standard veterinary bills will be less than $40? c. What percentage of standard veterinary bills will be between $80 and $150? d. What would be the veterinary bill amount that separates the cheapest 25% of visits for standard services? e. What is the range of costs for the middle 70% of standard visits? f. What is the IQR for the costs of standard veterinary services? 13. A college’s data about the incoming freshmen indicates that the mean of their high school GPAs is 3.4 with a standard deviation of 0.35. The distribution is normal. The students are randomly assigned to freshmen writing seminars in groups of 25. a. Describe the population distribution. b. Describe the sampling distribution for groups of 25 students. c. Find the probability a student has a GPA less than 3.2. d. Find the probability that one of the groups has an average GPA greater than 3.5. 6  Math-138 Fall 2024 14. The Center for Disease Control states that in 2017, 12.8% of American adults were smokers. a. Describe the sampling distribution for a sample of 50 adults. b. What is the probability that in a random group of 50 adults, more than 20% of them are smokers? c. Would it be unusual to select a sample of 50 adults and have less than 8% of them be smokers? Justify your answer using a probability. 7