Download Math 1105: Prelim 1 Practice Problems and more Exams Algebra in PDF only on Docsity!
Math 1105: Prelim 1 Practice Problems
General instructions: You do not need to simplify answers involving factorials, products, fractions, etc. You may consult the table of normal distribution values at the back of the exam.
- For each of the parts below, give examples of sets that satisfy the given conditions.
(a) Sets A, B and C so that A âȘ C = B âȘ C, but A 6 = B.
(b) Sets A and B so that A â© B = A.
- At the University of Mars, 12 of the incoming students take a course in agriculture, 107 of the students take a course in propulsion, and 15 of the students take agriculture but not propulsion.
(a) Whatâs the probability a randomly chosen incoming student is taking both courses?
(b) Whatâs the probability a randomly chosen student is taking either agricul- ture or propulsion?
(c) Given that a student is taking agriculture, whatâs the probability they are also taking propulsion?
(d) Are there any incoming students that are not taking either course? Briefly explain your answer.
- There are two vases. The first contains 3 balls labeled 1â3, and the second contains 4 balls labeled 1â4.
(a) Write down a sample space S for the experiment where you choose one ball from the first vase and one ball from the second.
(b) Let x be the sum of the numbered labels on the two balls that you draw. Write down the probability distribution for the value of x.
(c) Compute the mean (expected value) of x.
(d) Now suppose someone picks one of the vases at random (probability 1/ 2 each) and draws a random ball out of that vase. Given that the ball has an even value, what is the probability the ball came from the second vase?
- Angelica is a Cornell student who comes from New York City. The day before
Thanksgiving, she plans to take the Campus to Campus bus to visit her family. Let S be the event that there is a snowstorm on the day she is traveling, and let L be the event that her bus arrives late in New York.
(a) What do you think is the relationship between P (L) and P (L | S)? (Are they the same? If not, which one is larger and why?)
(b) Repeat part (a) for P (L) and P (L â© S).
(c) Suppose there is a snowstorm on the day before Thanksgiving in one out of every ten years. When there is not a snowstorm, the Campus to Campus bus arrives late 40% of the time due to heavy traffic. When there is a snowstorm, the bus arrives late 90% of the time. Compute the overall probability that the bus arrives late on the day before Thanksgiving.
(d) If Angelicaâs bus arrives on time, what is the probability that there was a snowstorm that day?
- Upon arrival at a hospitalâs emergency room, patients are categorized ac- cording to their condition as critical, serious, or stable. In the past year,
- 10% of the emergency room patients were critical.
- 30% of the emergency room patients were serious.
- the rest of the patients were stable
- 40% of the critical patients died
- 10% of the serious patients died
- 1% of the stable patients died
(a) Given that a patient survived, whatâs the probability the patient was cate- gorized as serious upon arrival?
(b) Are the events âpatient was categorized as seriousâ and âpatient survivedâ independent? Justify your answer using the definition of independence.
- A strand of DNA is represented by a string of the letters A, G, T, and C, each representing a specific amino acid. For example, AGGTCTAGGA is a string, and AAGGGGTTCA is a different string. Both are strings of length 10.
(a) How many possibilities are there for a strand of length 4 that contains every letter?
recalled, the FDA found that 12% tested positive for the bacteria; these were distributed randomly throughout the samples.
Prior to the recall, your friend bought 10 patties for a cookout.
(a) What is the probability of having zero infected patties at the cookout?
(b) What is the probability of having exactly one infected patty at the cookout?
(c) What is the probability of having at least one infected patty at the cookout?
(d) What is the expected value of the number of infected patties at the cookout?
(e) Your friendâs evil twin also bought 10 patties for another cookout, which was held at the same time on the same day. What is the probability that one of the two cookouts had no infected patties while the other cookout had at least one infected patty?
- The weights of adult female Dalmatians (a type of dog) are normally dis- tributed with a mean of 50 pounds and a standard deviation of 3 pounds.
(a) What percentage of Dalmatians weigh between 50 and 56 pounds?
(b) The top 10% of Dalmatians are above what weight?
(c) The weights of adult female Boxers (another type of dog) are also normally distributed, but with a mean of 57.5 pounds with a standard deviation of 1. pounds. Marty owns a Dalmatian and a Boxer. The Dalmatian weighs 45 pounds and the Boxer weighs 52 pounds. Which dog is more underweight? Give an explanation for your answer.
