Mathematics 310, Winter, 2004
Introduction to Mathematical Reasoning

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Instructor: Dr. Virginia M. Warfield
Office: Padelford C-437
Office Hours: Mondays 12:30-1:30; Wednesdays 1-2
Email: warfield@math.washington.edu
Telephone: 543-7445

Final Superproblem Superset

Here are all of the problems and lists:

First the basic techniques and tools:

I plan to give you a certain number of problems which test the techniques and definitions that should be automatic to you at this point. In this category I would include almost anything in Chapter 2; inductive proofs of numerical formulas; injections, surjections, composition of functions; cardinality; conditional probability, independence, Bayes formula and expected value.

Next the problems from the book: 1.35, 1.39, 1.49, 2.21, 2.26, 2.42. 2.49, 3.15, 3.29, 3.50, 3.56, 4.23-4.25, 4.34, 4.45, 4.46, 4.42, 4.43, 9.17 and 9.34.

Then the homegrown ones: 1) Jane and her parents all have brown eyes, but her sister has blue eyes. Jane’s husband Joe also has blue eyes. Jane and Joe have two brown-eyed children. What is the probability that their third child will have brown eyes?

2) A cab was involved in a deadly hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are told that:     
a) 85% of the cabs in the city are Green and the rest are Blue
b) a witness identified the cab as Blue.
The court tested the reliability of the witness under the same circumstances that existed on the night of the accident, and concluded that the witness was correct in identifying the color of the cab 80% of the time.

What is the probability that the cab involved in the accident was Blue?

3) Every McDonalds Happy Meal contains one of five different toys. The toys are randomlly distributed amongst the meals. How many Happy Meals would you expect to have to buy in order to collect one of each toy?
[NOTE: You may use without re-proving it the fact that if |a| < 1, then 1+a+a^2 + a^3 + … = 1/(1-a). Anything else you use you must prove]

4) An itinerate shyster has a box containing two coins. One is loaded so that the probability that it will come up tails is .7 . The other has tails on both sides. He can feel the difference between the coins, so when "randomly" choosing one from his box, he actually chooses the two-tailed one .65 of the time. He then tosses whichever coin he chooses. If it comes up tails, he tosses it again. If it comes up heads he tosses the other coin.
He offers the following bet to all comers: "If the two tosses give the same result (either both are heads or both are tails) you pay me a dollar, but if they are different, I will pay you TWO dollars!!!" What is the expected value of this game to a person taking him up on the bet?

5) An off-beat philanthropist has mixed some unusual Santa Clauses in with the standard Santa collection. As a result, 23% of the Santas in downtown Seattle are handing out $1 bills and 7% are handing out $5 bills . (Ordinary Santas just give a pat on the head.) This makes them feel very cheery, so the $1 Santas spend 65% of their time saying "Ho ho ho" and the $5 Santas spend 95% of theirs "Ho ho ho"ing. Ordinary Santas only "Ho ho ho" 30% of the time. What is the expected profit from meeting up with a "Ho ho ho" type Santa?

6) Prove that the composition of two bijective functions is bijective.

Late homework policy

Much as I prefer to have you hand in your assignments on time, I would much prefer to have you hand them in late than not at all. I am therefore establishing the policy that I will correct any paper that comes in, and deduct 10 points per class meeting of lateness (i.e., due on Monday and arriving Wednesday loses 10 points, arriving Friday loses 20 points, etc.) Anythin else you use you must prove

EXAMS ARE AVAILABLE!

I have put the finals of all those who so requested outside my office door. Medain for the class was 87/125, or 70%.

Have a good spring break!