STA218 Assignment for Quiz 4

Quiz on Thursday Feb 8th at the beginning of class

For this quiz, you will be allowed a single 8.5 by 11 one-sided formula sheet and a calculator. So write your formula sheet and use it to do the homework.

Probability and conditional probability

1. A jar contains 3 red marbles and 2 blue marbles. We sample two with replacement. What is the probability that they are both the same colour? My answer is 13/25.
2. Answer the question above if the marbles are chosen without replacement. My answer is 2/5.
3. The following is based on my experience as a market researcher. Of the white collar employees in an advertising agency, 50% are alcoholics, 20% are addicted to cocaine, and 15% have both characteristics.
   1. If a single white collar employee is chosen at random, what is the probability that he or she will be either an alcoholic, a cocaine addict, or both? My answer is 0.55.
   2. If a single white collar employee is chosen at random, what is the probability that he or she will be neither an alcoholic nor a cocaine addict? My answer is 0.45.
   3. Are alcoholism and cocaine addiction independent in this population? Answer YES or NO and show your work. My answer is NO.
4. A jar contains a fair six-sided die and a fair coin; the faces of the die have numbers 1 through 6, and the coin has a 1 on one face and a 2 on the other face. You shake up the jar and dump out the coin and the die at the same time. What is the probability that they show the same number? My answer is 1/6.
5. In a chain of fast food restaurants, 20% have had labour disruptions (strikes, etc.) during the past year, and 10% have had lawsuits. Eighty percent have had neither problem.
   1. If a single restaurant is chosen at random, what is the probability it had both a labour disruption and a lawsuit? My answer is 0.1.
   2. Are labour disruptions and lawsuits independent? Answer YES or NO and show your work. My answer is NO.
   3. If a restaurant has experienced a labour disruption, what is the probability it has also experienced a lawsuit? My answer is 1/2.
   4. If a restaurant has experienced a lawsuit, what is the probability it has also experienced a labour disruption? My answer is 1.
6. A jar contains 3 red marbles and 2 blue marbles. We sample two marbles  without replacement. What is the probability that the first marble was red given that the second was blue? Show your work. My work starts with a tree diagram. My final answer is 3/4.
7. Of the cars in a used car lot, 30% are stolen and 40% have mechanical problems. However, only 10% of the stolen cars have mechanical problems. Given that a randomly chosen car has mechanical problems, what is the probability that it was stolen? Show your work. I would start by making a Venn diagram, and obtain the probability of the intersection with the multiplicative rule. My final answer is 3/40.
8. Potential employees are interviewed by one of two officers in your Human Relations department. Call these interviewers A and B. The probability that A will hire any given job applicant is 0.30, while the probability that B will hire is 0.50. Interviewer A works twice as fast, interviewing twice as many applicants as B.

   Letting H represent the event that a randomly chosen job applicant is Hired, make a Venn diagram (the text would call it a probability table) divided into four regions. Label each region with a probability. Show how you got the probabilities of the intersections.

   Given that an applicant was not hired, what is the probability that she was interviewed by A? The Venn diagram is part of your work for this question. Show the rest of your work. My final answer is 14/19.

9. Also do exercises 3.42 and 3.56 from the text. Assume the populations are so large that sampling with replacement and sampling without replacement are basically the same.

Expected value

Here are some additional problems on expected value.

1. In a lottery, one thousand tickets are sold for $1 each. The winner gets $100. What is the expected profit from buying a ticket? Is this game fair? The solution is in your lecture notes.
2. A jar has one ball labelled with the number 1, two balls labelled 2, three balls labelled 3, and 4 unlabelled balls. It costs $1 to play the following game. The balls are mixed, and one ball is drawn at random. If a numbered ball is drawn, you win the number of dollars shown. Do you want to play this game? Assume you are guided by expected profit. Answer YES or NO and show your work. I get an expected profit of 40 cents.
3. In the game above, how much should the operator of the game charge for playing if she wants to make it fair? My answer is $1.40, because the extra 40 cents cancells out the expected profit of the last question. Also, the expected profit is (1-D)*.1 + (2-D)*.2 + (3-D)*.3 +(0-D)*.4, where D is the amount you pay to play the game.
4. In Exercise 3.36, assume the premium is $3000. What is the company's expected profit? My answer is $950, because with a premium of D, the expected profit is D*.94 + (D-80000)*.01 + (D-25000)*.05.
5. In Exercise 3.36, assume the premium is $800. What is the company's expected profit? My answer is $98, because with a premium of D, expected profit is .85*D + .8*.15*(D-3000) + .12*.15*(D-9000) + .08*.15*(D-15000).

Do the assigned problems in preparation for the quiz. They are not to be handed in.