Math 425, Fall 1997

Prof. Hochster

During the semester I will assign, sporadically, thirty or more optional problems designated ``extra credit problems." Many of them will have a specified deadline for handing in solutions. Others, which are intended to stimulate thinking about the material, may have open deadlines for a while. Some of these problems will be more challenging than ordinary homework problems, but a substantial number will be no harder. In some cases they will be used to develop important material for which there is not enough time in the regular lectures.

Do not turn in extra credit problems with regular homework: give them directly to me. Each problem will receive one of three grades: check+, check, or check- . The first (check+) means that the solution was essentially correct. The second (check) means that there was substantial insight into the problem but that there was still a major component missing in the solution. The third (check-) recognizes that some effort was made, but indicates that the solution was too far from correct for substantial credit.

I will use these problems in grading in several ways: for any two check+'s or any four check's I will replace one of your lowest (or missing) homework grades with a perfect score. If you are close to but beneath a grade line I will be greatly influenced in deciding whether to give you the higher grade by performance on extra credit problems. However, I am hoping that people will try these problems not for the possible effect on their grades, but rather for challenge and enjoyment.

** EC#1.** You are a contestant on a quiz show. There is a wonderful
prize behind one of three doors --- behind the other two, nothing.
You choose one of the doors. The quiz show host, who knows where the
prize is, opens one of the wrong doors. You are then given a choice:
you can get what is behind the door you chose originally, or you
can switch to the other unopened door. What should you do?
More precisely, what is the probability that you will win the prize if
you stick with your original choice? If you switch?
[9/10]

** EC#2.** What happens in the preceding problem if there are seven
doors instead of three (still just one prize), and the host opens five
wrong doors before giving you the choice of switching? What if there
are n doors, where n is an integer at least 3, one prize,
and the host opens n-2 wrong doors? [9/12]

** EC#3.** What is the probability that at least two students in a class of
30 will have the same birthday? For simplicity, assume that no one in the
class was born on February 29, and that all other dates are equally likely.
[9/15]

** EC#4.** In a certain country the percentages of boy babies
and girl babies born are equal under normal circumstances, and their
life expectancies are the same. But a tradition
develops that families keep having children until a girl is born,
and then stop. (Of course, they may have to stop for other reasons.)
What is the expected ratio of males to females in the
population? [No deadline yet]