Square of Dierences:
Experimentation, Conjecture, and Proof
The Game
Mathematics involves experimentation and discovery, generalization and conjecture, and proof. The game Square of Dierences supplies a valuable exercise in searching for understanding. Th
WHAT IS A
BIOSTATISTICIAN TRYING
TO PROVE?
PROOF IN MEDICAL RESEARCH
Susan S. Ellenberg, Ph.D.
Department of Biostatistics and
Epidemiology
Perelman School of Medicine, U
Penn
Statistics is the science of uncertainty
Biostatistics is the application of
LAW 520 & MATH 220 & PHIL 220
Problem Set 3
Fall Term, 2012
Due in class on Tuesday, November 27 or in my mail slot in 433 Cohen Hall by 1:00 p.m.
the same day. Complete four of the following problems.
1. Let D be the statement (S (S (0) = S (S (S (S (0).
Markov Chains: How Google Works
Example 1 Say every year 5% of the people in California move Elsewhere while 1% of the people livine Elsewhere move to California. Is
there an equilibrium population distribution, and if so, what is it?
Let C0 be the initia
Math/Philosophy 220, Law 520 Fall 2012
Problem Set 1
Due:
In class Thursday, Sept. 20. Late papers will be accepted until 1:00 PM Friday.
Remark:
Suggestion for #4b).
4. The number e is dened as e = 1 + 1 + + + + . Prove that
number by the following steps
Math/Philosophy 220, Law 520 Fall 2012
Problem Set 1
Due: In class Thursday, Sept. 20. Late papers will be accepted until 1:00 PM Friday.
Remark: It is often eective to work together with someone else.
1. Comment on the following proof that there is no un
Five Men
There are ve houses in a row on a street. In each lives
a man of a dierent nationality, each preferring a dierent beverage, driving a dierent make of car, and having
dierent jobs.
1. The Englishman lives in the red house.
2. The Egyptian is a sal
Cancer Test
100
cancer
100%
0%
100
test positive
No false negative.
0
test negative
1% have cancer
10,000
people
99%
21%
9,900
no cancer
2,079
test positive
79%
7,821
test negative
Probability (cancer | test positive) =
The test is 79% accurate,
so 21% fa
Baysian Probability Problems
1. [From Thinking Fast and Slow by D. Kahneman, Chapter 16]
A cab was involved in a hit-and run accident at night.
Two cab companies, the Green and the Blue, operate in the city.
Data:
85% of the cabs in the city are Green, 1
LAW 520/MATH 220/PHIL 220 Fall 2012
1
1
Memoir 12.11.08
One of our goals in this section of the course is to understand the extent to
which philosophical reection on the nature of mathematical knowledge and
mathematical truth has been shaped by the develo
Proof in Mathematics, Philosophy, and Law
Some Topics
Proof: Convince a dubious person. It depends on the audience.
1. I have a neck, a bottle has a neck, therefore I am a bottle.
2. Proofs Without Understanding
Let N := 15 7 8 23 41.
Is N divisible by 10