Click here to toggle music on and off – Enya’s “Storms in Africa”


The following discussions on mathematics are offered for your edification and asmuement. Note some of the information on multi-dimensional geometry is required reading for some of the discussions in the corresponding PHYSICS BLOG.

Primer on Multi-dimensional Geometry

Why do we need to study statistics?
Math Jokes
How to play 3 and 4 dimensional tic-tac-toe
Behavioral aspects of metrics - Tom Demarco
Gambler's remorse
Why do they teach binary in computer classes?
Signary math
Conditional probability - Bayesean statistics

========

The Math Blog

Why do we need to study statistics?

The main reason that we need to study statistics is that we are, by nature, very poor judges of probability. ...

Here are some excerpts of a short seminar I have demonstrating why we need to study statistics

EXAMPLE 1 - Medical tests
A man takes and annual physical and takes a routine blood test.
The results of one test (which is 99% accurate) indicates that he has a rare disease (only 1 in 10,000 people have this condition) which is always fatal.  If the test is true he has less than one week to live.
How worried should this man be?

Click here to review the math

EXAMPLE 2 - Buying a fuel efficient car
You have two cars. 
One is  4 door sedan that gets 30 mpg
The other is a rugged truck that gets 12mpg
You can buy either

  • A new SUV that gets 18mpg to replace your truck (6 mpg improvement)
  • Or a new hybrid that gets 50mpg to replace your sedan (a 20 mpg improvement)

Which is the best choice? Upgrade the truck (improving 6mpg)? or upgrading the sedan (improving 20mpg)?
Which saves the most gas?

Click here to review the math

EXAMPLE 3 - Buying stocks
You are considering buying one of two stocks
Stock A is volatile and over past years has consistently dropped 50% and then rebounded with an 80% increase.
Stock B is steady and consistently gains 5% per year
Assume that past performance will be repeated. Which stock should you buy?

EXAMPLE 4 - Inferences
Read the following paragraph and rank order the following statements from the most probable to the least probable (i.e. with the highest probabilities at the top, Number 1)

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

A. Linda is a Democrat
B. Linda is a bank teller
C. Linda is a member of the League of Women Voters
D. Linda works in a bookstore and takes Yoga classes
E. Linda is a teacher in elementary school
F. Linda is in the feminst movement
G. Linda is an insurance salesperson
H. Linda is a bank teller and is active in the feminist movement

Click here to review the math

EXAMPLE 5 - Monti Hall - Let's make a Deal"

Two decades ago there was a game show called “Let’s make a Deal”. The final game each week involved selecting one of three prizes hidden behind large doors. One would have an expensive prize like a Hawaiian vacation or a car; the other two doors would be gag prizes like two dirty goats.

If you select one door, you have a 1/3  chance of winning the car. But after you pick a door, the host would always open one of the other doors and show you a goat and he would give you the opportunity to chance your choice and select the other unopened door.

Question? Should you keep your current door? Change? Or doesn’t it matter (meaning the odds of either unopened door are the same)

Click here to review the math

EXAMPLE 6 - OJ Simpson trial

During the OJ Simpson trail in the early 1990's the prosecution endeavored to prove that OJ was an abusive husband. They produced witnesses, and photographs, and hospital reports and police reports and xrays showing how frequently Nicole Simpson was beaten and bloodied at hom eand in public places like restaurants and movies.
Eventually the defesne team admitted that OJ was an abusive spouse, but they offered an interesting statistic.

They claimed that there are nearly 2,000,000 abused women but that only about 1,000 of these women are killed. Therefore being an abusive husband would only mean that there is a 1 in 20,000 chance that OJ was a killer.

Is this a good statistic?

Click here to review the math

      Return to Top

--------

Math Jokes

There are liars, damned liars, politicians, and statisticians.

Did you hear about the woman who went into a pizza parlor and ordered a pizza. The attendant asked if she wanted the pizza cut into six pieces or eight. The woman replied, "Oh, just six. I couldn't eat eight."

There are 10 kinds of people in the world. Those that know binary and those that don't

Who can count the cost of innumeracy?

Did you hear about the two statisticians who got married? When they made their vows they "failed to reject" each other.

There are three kinds of people in the world; those who can count and those who can't.

47.9% of all statistics are made up on the spot.

99% of all lawyers give the other percent a bad name.

Every three seconds a woman somewhere in the world has a baby. That woman must be found and stopped!

Old mathematicians never die; they just lose some of their functions. 

A statistician is someone who is good with numbers but lacks the personality to be an accountant.

Philosophy is a game with objectives and no rules. Mathematics is a game with rules and no objectives. 

God is real, unless proclaimed integer

Q: Did you hear the one about the statistician?
A: Probably.... 

Three out of every two people have trouble with fractions

1 + 1 = 3, for large values of 1

Life is complex: it has both real and imaginary components.

----------

Question: "How many seconds are there in a year?"Answer: "Twelve. January second, February second, March second, ..."

----------

 

----------

Q: Did you hear that joke about the infinite line?
A: Don’t worry, It doesn’t have a point!

----------

Why are 2011 quarters more valuable than 2001 quarters?
(Mouseover the text in the region below to see the answer)

      Return to Top

--------

How to play 3 and 4 dimensional tic-tac-toe


      Return to Top

--------

Behavioral aspects of metrics - Tom DeMarco


      Return to Top


--------

Gambler's remorse


      Return to Top

--------

Why do they teach binary in computer classes?


      Return to Top

--------

Signary math

This is a bizarre topic.  It is about a different numbering system than you have probably ever heard of before.  It is Signary math.

There are a number of different numbering systees that you may become familiar with going through school. The decimal system (base 10) is the most common. But programmers and computer practitioners may become exposed to others, such as:

Binary (base 2)
Octal (base 8)
Hexidecimal (base 16)

Then there are historical variants.

Some early civilizations used base 12 numbering systems, duodecimal systems.
Vestiges of these counting systems explain why there are 12 hours in a day, twelve items in a dozen, twelve inches in a foot, twelve dozens in a gross, etc.

Early Bretons, Danish and early French used a “Vigesimal”, base 20 numbering system.  Remnants of those counting methods remain embedded in those languages to this day. (For instance the number 80 is French is quatre-vingts, the French word for 80, literally means "four twenties”). Base 20 systems were also used by the Albanians, and the Basque, Irish gaelic and Welsh.

The Babyloneans used a base 60 numbering system, a “Sexagesimal” system, (That is why there are 60 minutes in an hour, and 360 degrees in a circle and so on) primarily because they did not have a demical point and they had difficulty with fractions to a base 60 system allowed a lot of division to be reduced to integers (1,2,4, 5,6,10, 12,15,20,30 are all integral fractions of 60)

But there is a new mode of math I want to tell you about. (Well not new.  It gained brief popularity a few years ago but has since been all but forgotten. I have been unable to trace its origins except for a few apocryphal stories that I will not share here)

It is “signary” math.  It is a base 3 numbering system (ternary), but different from the normal base 3 math you math have touched on briefly in high school math classes.  It is not ternary math, per se, although there are three symbols.

With binary math you have two symbols (0 and 1) and count thus

0001    1
0010    2
0011    3
0100    4
0101    5
0110    6
0111    7
etc.

In base-3  math, or Ternary math, you have three symbols, 0,1,2 and count.

0001    1
0002    2
0010    3
0011    4
0012    5
0100    6
0101    7
0102    8
0110    9
etc.

In ternary math you have three symbols (-,0,+) - They are a PLUS SIGN, a ZERO and NEGATIVE OR MINUS SIGN.

These symbols affect the place value just like the normal symbols 0,1,2 but ...watch

0000    0
000+    1 (that is one 1)
00+ -    2 (that is...3 minus1)
00+0    3 (that is 3)
00++    4 (that is 3 plus 1)
0+ - -   5 (that is 9 minus 3 minus 1)
0+ - 0   6 (one 9 minus one 3)
0+ -+   7 (9 minus 3 plus 1)
0+0 -    8 (9 minus 1)
0+00    9 (one 9)
0+0+    10 (9 plus 1)
0++ -   11 (9 plus 3 minus 1)
0++0    12 (9 plus 3)

Each number is a small equations. But then this is true with decimal math as well. For instance, 13 means one ten plus three ones.

Arithmetic operations in singary are different... actually they are quite odd.
For isntance, turning positive numbers into negative ones is done without leading signs.
Instead you simply change all the plus symbols in a number to minus signs and all the minus signs to plus signs.

1999 would be   +0 -+ -00+

-1999 would be  -0+ -+00 –

Multiplication and division but some numbers becomes trivial. For instance, when you multiply by three you just shift the numbers to the left and add a zero.

10 is      0+0+
30 is     0+0+0

… just like multiplying by ten in a decimal system.

Similarly, dividing by three is trivially easy, You just shift the symbols to the right.

Where binary numbering systems model two states (on and off) and correspond to binary or Boolen Logic (true/false), there are a few places three base states would be useful or where such a numbering system might work well.

For instance this would be the ideal numbering scheme for optical computers that have, say, three physical state values (left polarized, right polarized and non polarized light)

There are a few other places where signary math might be well suited. In programming for instance, were values might be evaluated upon ternary conditons such as (LESS THAN; EQUAL;GREATER THAN) but such applications are not common and are typically addressed with sequential or nested binary logic statements.

Anyway, it’s just an oddity for those who are seriously math enthusiasts.  If you are one ...have fun!

      Return to Top

--------

Conditional probability - Bayesean statistics

A Drunkard's Walk - How Randomness affects our lives

Examples in example 6 above (WHY STUDY STSTICTICS)

 

      Return to Top

========================

Answers to the WHY STUDY STATISTICS eamples given previously

EXAMPLE 1 -MEDICAL TEST
Consider this test being performed on 1,000,000 people
Only one in 10,000 people would have the disease. That would be 100 of the million are really sick. The other 999,900 are healthy
Now calculate the results of a test that is 99% accurate on these two groups

Because there are so many more negatives than positives, there will be many more false positives than true positives.

Return to Top

EXAMPLE 2 - BUYING A FUEL EFFICIENT CAR

  1. Truck 12 mpg equates to 83.3 gallons / 1000 miles
  2. SUV (18 mpg) uses 55.5 gal/1000 miles
  3. Sedan (30mpg) uses 33.3 gal / 1000 miles
  4. Hybrid (50 mpg) uses 20.0 gal / 1000 miles
  5. Upgrading your truck from 12 mpg to 18 saves 26 gal
  6. Upgrading your sedan from 30 mpg to 50 saves 13.3 gal

Return to Top

EXAMPLE 3 - BUYING THE BEST STOCK

If you had $200 in shares of both A and B what would have happened last year?
Stock A

$200 – 50% = $100 + 80% = $180 (net 10% loss)

Stock B

$200 + 5% = $210 (net 5% increase)

Which stock should you have bought?
Stock B is the better stock. Stock A is a consistent loser

Return to Top.

EXAMPLE 4 - Inferences

The trick to this question is that people dramatically mis-interpret porbabilities

A. Linda is a Democrat
B. Linda is a bank teller
C. Linda is a member of the League of Women Voters
D Linda works in a bookstore and takes Yoga classes
E. Linda is a teacher in elementary school
F. Linda is in the feminst movement
G. Linda is an insurance salesperson
H. Linda is a bank teller and is active in the feminist movement

Nearly 90% of people, for instance, will list H higher than B or F

Consider that if 10% of people are bank tellers, not all of them will be feminists. Therefore H must be lower than B.

However, because since both B and F are deemed likely by most readers, perceptuallymost people mentally add the two likelihoods together, rather than noting that the combination must be smaller than either.

 

Return to Top

EXAMPLE 5 - Monte hall – double your chances?

When you first select a door, all are closed and the odds of picking the car is 1 out of 3 or 1/3 or 33.3%
But when the host opens a door showing goats, a non-random act has occurred. 

  • If you keep your current choice, you have a 33.3% chance of winning a car.
  • If you switch doors and take the OTHER unopened one, your chances are doubled; you have a 66.6% chance of winning

You should definitely switch to the other unopened door!


      Return to Top

EXAMPLE 6 - OJ Simpson Trail

The statistic that the odds of an abusive husband becoming a murderer is correct. It is 1 in 20,000

However, there is a fallacy here. It is so common that it is known as the prosecutor's fallacy. This statistic of 1 in 20,000 is being misused. Essentially, you are saying that given that OJ is an abusive husband, the odds of nicole Simpson dying are 1 in 20,000.

But wait! Nicole Simpson IS dead. So you should only look at the 1,000 women who die each year. (This is called conditional probability or Bayesean statistics)

So given that an abused wife has been killed, the question you should ask is "Of the 1,000 abused women killed each year, how many were killed by their abusive husbands?"

For the year 1993, the year of OJ's trial, the answer was 90%.