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STAT 3000
Homework assignment 5
Due Tuesday, February 13, 2001
NAME: __________________________
Work on the homework individually or in groups (recommended - but remember: every person has to turn in his/her individual assignment sheet). Show your work as much as possible (where applicable). Answers without work shown are not acceptable. Number of points you get for each problem is given in parentheses. You can get a maximum of 30 points for this homework.
I Textbook Problems on Probability (17 pts)
Please work on the following problems from Chapter 1of the Hayter book:
1.5.7 (2 pts), 1.5.11 (2 pts), 1.5.13 (2 pts - see Assignment #1, Problem 6a) 1.6.1 (2 pts), 1.6.4 (2 pts) 1.7.1 (1 pt), 1.7.2 (1 pt), 1.7.3 (1 pt) 1.7.4 (2 pts), 1.7.8 (2 pts)
For problems 1.7.1, 1.7.2, and 1.7.3, do parts (a) and (c) "by hand" and use your calculator or an appropriate Excel function for parts (b) and (d).
II The story of a cheating student (6 pts)
Instead of studying for the exam, one STAT 3000 student (out of a class of 46 students) decided to trust his neighbors' solutions. Unfortunately, the student did not know that the question has been formulated in such way that only 4 out of 45 his classmates will answer the question correctly, while all other 41 his classmates will answer the question incorrectly. In the classroom, all students have been randomly assigned to a seat and our particular student does not know anything about the performance of his/her neighbors.
a) (1 pt) In a first attempt, our student takes the following strategy #1: he/she decides to look at his/her left neighbor's exam and copy the solution from this neighbor. What is
the probability that the student will answer the question correctly? (Note: to be able to answer part c) below, you should work with fractions, e.g. 1/3 rather than with decimals, e.g. 0.333).
b) (4 pts) Not really satisfied with the chances of answering the question correctly when following the strategy #1, the student decides on the following strategy #2 in order to increase his chances of answering the question correctly: look at his/her left and right neighbors' solutions. If these are identical, then copy (either) solution. If they are different, then toss a coin to decide whose solution to copy. We can assume that the 4 correct answers are identical and all 41 incorrect answers are identical as well (but, of course, different from the correct answers). What is the probability that the student using this strategy #2 will answer the question correctly? A probability tree might help you answer this question.
c) (1 pt) So, based on your results, what strategy would you recommend to achieve best results on the exam?
III Let's Make a Deal (The Monty Hall Problem) - 7 pts
The Monty Hall Problem gets its name from the TV game show, "Let's Make A Deal," hosted by Monty Hall. The scenario is such: you are given the opportunity to select one closed door of three, behind one of which there is a prize. The other two doors hide "goats" (or some other such "non-prize"), or nothing at all. Once you have made your selection, Monty Hall will open one of the remaining doors, revealing that it does not contain the prize. He then asks you if you would like to switch your selection to the other unopened door, or stay with your original choice.
Here is the problem: Which strategy maximizes your chances of winning?
(i) Stay with your first selected door; (ii) It does not matter whether you switch or not - the chance of winning is always the same; (iii) Switch to the other (unopened) door.
a) (1 pt) By intuition, what would be your answer?
b) (3 pts) Now, let's play a little game to check your answer. Here is a "Let's Make a Deal" applet that provides a simulation of the Monty Hall problem:
http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html
