PURPOSE OF SEMINAR........................................................................................................................................... 2
HISTORY OF PROBABILITY THEORY ........................................................................................................................ 2
SURPRISING RESULTS OF PROBABILITY .................................................................................................................. 3
M ONTY H ALL P ROBLEM .................................................................................................................................................. 3
B IRTHDAY PROBLEM ........................................................................................................................................................ 5
100 C OIN F LIPS ............................................................................................................................................................. 5
D OUBLE D ECKER ............................................................................................................................................................ 6
BAYES THEOREM ................................................................................................................................................... 6
F ALSE P OSITIVE HIV TESTS ............................................................................................................................................... 6
O.J.
S IMPSON C ASE ........................................................................................................................................................ 7
BENFORD’S LAW .................................................................................................................................................... 8
C
ORPORATE
F
RAUD
........................................................................................................................................................ 8
TAKING ADVANTAGE OF PROBABILITIES ................................................................................................................ 8
C RAZY D ICE ................................................................................................................................................................... 9
R OULETTE W HEEL .......................................................................................................................................................... 9
V IRGINIA L OTTERY ........................................................................................................................................................ 10
GAME THEORY ..................................................................................................................................................... 10
C LOWN W AR .............................................................................................................................................................. 10
TOO GOOD TO BE TRUE ....................................................................................................................................... 11
S TOCK S CAM ............................................................................................................................................................... 11
N O W HAMMIES ! ......................................................................................................................................................... 12
M ILLIONAIRE S CANDAL .................................................................................................................................................. 13
NOT SO SURPRISING COINCIDENCES .................................................................................................................... 14
S TOCK M ARKET G ENIUS ................................................................................................................................................ 14
E XPECTING THE I MPROBABLE .......................................................................................................................................... 15
ORDER IN CHAOS ................................................................................................................................................. 15
B
UFFON
’
S
N
EEDLE
........................................................................................................................................................ 15
D
RIVING
C
ONTEST
........................................................................................................................................................ 16
CONCLUSION ....................................................................................................................................................... 17
SELECTED BIBLIOGRAPHY..................................................................................................................................... 17


Probability is so prevalent in our culture

Probability sharpens the mind

Increase number and probability literacy in our students
"Innumeracy, an inability to deal comfortably with the fundamental notions of number and chance, plagues far too many otherwise knowledgeable citizens." – John Allen Paulos

Unveil the order amongst the chaos

Cultivate a joy of mathematics

Modern probability theory has its roots in some of the early mathematicians of the renaissance.

Gerolamo Cardano, also famous for his role in the great story of a cubic solution, was a famous physician, mathematician and a gambler.
His passion for gambling and interest in mathematics led him into giving the first systematic treatment of probability

Pierre de Fermat and Blaise Pascal, two other famous Renaissance mathematicians, were also critical in the development of probability theory.
Their involvement started when Fermat received a letter from Antoine Gombaud

Gombaud was an amateur mathematician and also interested in gambling starting to think about questions of probability

One game he employed where he won more than he lost was where he bet that he could roll a 6 in four rolls
Why did Gombaud win at this game?
Probability in the long run was on his side
How can we determine this probability?
One good strategy to determining probabilities is to look at the opposite (similar to Euclid’s
Reductio Ad absurdum or proof by contradiction)
What is the probability of him NOT rolling a 6 on the first roll o 5/6… how about on the 2 nd , 3 rd and 4 th rolls o 5/6 x 5/6 x 5/6 x 5/6 = 625/1296 = .482 o 48% of NOT rolling a 6 therefore… o 52% chance of getting a 6 in four rolls

He then developed a second game he thought would give similar results to the first
Rolling two dice, he bet he would roll a 12 in 24 rolls
BUT, he started to lose money

Why?
Again, employing strategy 2, what are the chances that he doesn’t roll a 12?
35/36… how about 2 nd , 3 rd , 4 th … 24 th
35/36 x 35/36 x 35/36…. (35/36) 24 = .51 = 51% chance of losing therefore…
49% chance of winning
He wrote to Blaise Pascal asking for help in solving the problem, Pascal would write to
Fermat (and the great Father Mersenne was involved too) and this correspondence laid much of the foundation for modern day probability theory
“Probability theory is nothing but common sense reduced to calculation.” – Pierre-Simon Laplace

Christian Huygens, a teacher of the great Leibniz, learned of the correspondence of Pascal and

Fermat and wrote the first book on probability theory
Galileo, Rene Descartes, and the Bernoulli family also had an impact on probability theory
Nowhere does our intuition fail us more than probability… there are so many results that are counterintuitive to our human logic.
Monty Hall Problem

At the beginning of my math classes, every Tuesday, we have an exercise called “Pick your Poison”

One student in my class, each Tuesday gets to pick one of 5 games to play against Mr. Edwards

If they win the game, they get a matchbox car and if they lose, I give them a piece of candy, winner either way but they obviously want the car more

One of the five games I call Boxed Car Challenge, this game I played with [name of participant] over here, is basically similar to what is known as the “Monty Hall Problem”

In a 1990 “Ask Marilyn” column in Parade magazine the Monty Hall Problem was posed… o First, Marilyn is Marilyn vos Savant, a woman famous for having one of the highest recorded IQ’s in history o The problem sent to Marilyn was the following…
“Suppose you're on a game show, and you're given the choice of three doors:
Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say
No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?"
Is it to your advantage to switch your choice?” o Such a situation is very similar to the actual game show Let’s Make a Deal hosted by
Monty Hall and thus the name “the Monty Hall problem”

What do you think?

Should you keep your original answer or is it advantageous to switch? o Most people think the chances are 50/50 either way and there is no difference o But…. Let’s take a look


If you pick door #1, your chances are 1/3 that it is behind that door… what are the chances that it is behind door #2 or door #3… 2/3

Now when the host opens up one of the doors, let’s say door #2, he is helping you, he is eliminating one of the possible wrong answers… what is the possibility that the car is behind door #3 now? o Well, it’s still the same as the original 2/3 so it is to the person’s advantage to switch

Another way to look at this is by exaggerating the situation
One way to understand this situation is to exaggerate the situation, which is a helpful method in many areas of mathematics.
Let’s exaggerate the game show to have 1,000 doors.
Monty Hall asks you to choose one of the 1,000. Are you expecting to win the car? No, of course not.
Assume you pick door 76.
Let’s say the contents of 998 doors were then revealed all without the car. You are now left with the closed doors 76 (your original choice) and door 624.
Monty Hall asks you if you would like to switch. What do you think? Should you switch?
Of course.
Your chance of getting the right door on the first guess was 1 of 1,000.
Once all the doors have been eliminated but one, if you switch your probability would then switch to 999 of 1,000. I’ll go for the switch.

The answer to this problem is so counterintuitive that vos Savant received 10,000 letters claiming she was wrong and 1,000 were from Ph.D’s
“Since you seem to enjoy coming straight to the point, I'll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and in the future being more careful.”
Robert Sachs, Ph.D.
George Mason University
“You blew it…”
Scott Smith, Ph.D.
University of Florida
“May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?”
Charles Reid, Ph.D.
University of Florida
“Maybe women look at math problems differently than men.”
Don Edwards
Sunriver, Oregon

Even the great Paul Erdos got the problem wrong which gives me great satisfaction because this gives me some comfort for the many dumb mistakes I make in my class room.


If you still don’t believe me, I encourage to test this scientifically (not mathematically) by simulating the situation. I have done so twice and both times my results coincided with the results above
Birthday problem

How many people would there have to be in a group in order for the probability to be half that at least two people in it have the same birthday? o Contrary to our intuition the number needed is not nearly as high as we think… in a class of
22 students and 1 teacher (23 total)… it is more likely than not at 51% (fun to illustrate to classes of 23 or more) o Try this with your classes o A person tried to explain this once on the Johnny Carson show and Carson didn’t believe it… he asked how many them shared his birthday of October 23 rd and none raised their hand… the non-mathematician guest was unable to respond (more innumeracy)… the probability doesn’t state that there is a 50% chance of any birthday (say October 23), it says that there is a 50% of a birthday in common
100 Coin Flips

One interesting experiment that can be done with your students is a coin flipping exercise

Let’s say you have a class of 20 students

Tell the class that they are doing an experiment where the teacher will leave the room, and the students are to flip a coin 100 times and record the results on a piece of paper

However, there is a twist… 10 students actually flip a coin, and the other 10 “mentally” flip a coin or basically fake the results

You tell the students that you will come back in and attempt to correctly determine who faked and actually flipped by looking at their results… if they stump you, they get a piece of candy

You can come back in and almost certainly by looking at the results determine the students that actually flipped with an understanding of probability
Example of 100 coin flips (o represents heads, x represents tails)
O
O
O
O
O
X
O
X
O
X
X
X
O
X
O
O
X
O
O
O
X
O
O
O
X
O
X
X
X
O
X
X
O
X
X
X
O
X
O
X
O O O X O O X O X O
O
O
O
O
O
O
X
O
O
O
X
O
O
X
O
X
X
X
X
X
O
O
O
O
O
O
X
O
X
X
X
X
O
X
X
X
O
X
X
O

X O O X X X O O X O

Most human minds don’t appreciate the occurrence of long “runs” when flipping a coin 100 times o Almost every set of 100 tosses of a coin will have a run of 5 heads or tails (or more) o 100 flips will usually generate around 3 such strings o the above random selection generated 2 strings of 5 heads

Therefore, you take the results and look for at least one run of 5 heads or tails and these are the genuine flippers

Related to this, although flipping a penny has a probability of 50/50 (equal) o If you were to stand 100 pennies up on their side, slam a table, the probability is actually more likely to land heads o If you were to spin a penny over and over again, the probability is more likely that the penny will land tails
Double Decker

This game, I call it double decker is another game that I play with my students in the Pick your
Poison exercise

We think it would be unlikely in 52 flips of the two decks of cards to get the exact same card at the same time

However, you have a greater than 60% chance of this happening

Thomas Bayes, was an English mathematician and Presbyterian minister in London in the early 18 th century

He is most famous for Bayes’ Theorem, a concept important to probability theory

Bayes’ Theorem is rooted in what is known as conditional probability, the concept that probability changes when outcomes of particular events are connected o For example, the probability that a randomly chosen person knows how to find derivatives of logarithms and the probability that a randomly chosen person is a Calculus teacher are both unlikely BUT… o The probability that a person knows how to find a derivative of a logarithm if they are a
Calculus teacher is much higher

Bayesian analysis occurs often today in every day occurrences

However, many mistakes are made in the application of Bayes’ Theorem
False Positive HIV tests

One place we see Bayes’ Theorem abused is in the medical profession

Let’s say that you applied for life insurance and the insurance company required you to get a blood test

Let’s say your doctor then called you up and informed you that the blood test came back testing positive for HIV


You then would likely ask, how sure are you?

Only 1 out of 1,000 tests is a false positive so it is almost certain that you have HIV

Bayes’ Theorem argues the contrary

If a random person were to enter a hospital today and be tested for HIV and received a positive, it is more likely they do not have HIV

Let’s employ Bayes’ method o 1 st – define the sample space… statistics from the CDC report that about 1 in 10,000 heterosexual American who get tested were truly infected with HIV
 This means that if 10,000 people were to randomly walk into a hospital, only one would TRULY have HIV o 2 nd – define the members of the space… this is the number of false positive which we have already said is 1 in 1,000
 This means that if 10,000 people were to randomly walk into a hospital, 10 people will test positive when they really do not have HIV o Let’s put these two together… 10,000 random people (not drug abusers) walk into the hospital, 10 are false positives and 1 is tested who truly has it so o 11 total are told they have HIV but in actuality only 1 really has it

Simplified, if a random person were told he/she had HIV, the probability is 10/11 the he/she does not have it o Obviously this changes if the person is not a part of the sample space, if they have a history of drug abuse, are not heterosexual etc.

Some might think this is not that big of a deal, when in actuality false positive HIV tests, false positive mammograms, etc. have had an adverse effect on many people and actually led some people to attempt suicide when they did not even have the disease
O.J. Simpson Case

Another instance where Bayes’ theorem was employed incorrectly was in the O.J. Simpson case

The defense (Dershowitz) for Simpson employed the following argument defending Simpson o 4 million women are battered annually by husbands and boyfriends o However, according to FBI Crime reports, a total of 1,432 were killed by their husbands o Few people who abuse their wives go on to murder them… o Is this true? Yes. o Relevant? No. o It is how the question is phrased whether we are determining probability correctly o Dershowitz asked the question… What is the probability that a man who batters his wife will go on to kill her? o The right question is… What is the probability that a battered wife who was murdered was murdered by her husband?
 This statistic is very incriminating, the FBI reported that in 1993, 90% of battered women that were killed were killed by their abuser


Benford’s Law was discovered by an American astronomer named Simon Newcomb around 1881

He found, when looking at the pages of books of logarithms, that the pages beginning with the number 1 displayed more “wear and tear” than numbers beginning with 2 to 9.

Ultimately it was observed that in lists of numbers of real life data… for example, if you were to list all the numbers found in a Sunday edition newspaper, a majority of those numbers would likely start with 1 o A corollary to this is that 2 is more common than 3 and 3 more common than 4 and so on with 9 being the least common initial digit… the probabilities for infinitely large lists of numbers tends towards the following;
Initial Digit Probability
1 30.1%
2
3
17.6%
12.5%
4
5
6
7
8
9
9.7%
7.9%
6.7%
5.8%
5.1%
4.6% o It has been found to be true for electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, etc.
Corporate Fraud

Benford’s Law has been used often to find corporate accounting fraud and other related scams that involve lists of numbers

One instance involved a young entrepreneur named Kevin Lawrence, who raised $91 million to start a chain of high-tech health clubs o He and some of his cohorts ended up taking much of the invested money and investing it in personal watercraft, 47 cars (Hummers, Ferraris), Rolex watches, a $200,000 samurai sword, a cotton candy machine and other let’s say unnecessary items o To cover up the spending spree, they were forced to invent numbers and they evenly distributed the numbers between the 9 digits o A quick analysis of Benford’s Law was able to show a greater likelihood of corruption o Lawrence was sentenced to 20 years in prison
“Chance favors only the prepared mind.” Louis Pasteur

Crazy Dice
(Non-transitive Dice)

In the dice game I played at the beginning with [insert name of participant here], we would think that being able to choose first is better

However, once again, in probability, the truth is contrary

Let’s see why you should want to choose 2 nd

Take a look at each die individually and compare them

The red die has 5 4’s and a 1.

While the blue die has 5 3’s and a 6.

And the green die has 3 5’s and 3 2’s.

We can see here that the red die with 5 4’s is better than the blue die with 5 3’s (the red wins with a probability of 25/36)

The blue die is better than the green die with a 21/36 probability

So, we are tempted to think the red is the best BUT look at the blue in connection to the red

The blue is better than the red with a 21/36 probability

It is circular and each one beats another thus making the best die the one chosen 2 nd (assuming you choose wisely)

These are called Non-transitive dice and can be bought online at a neat website called grandillusions.com
Roulette Wheel

According to legend, the Roulette wheel was invented by the great Blaise Pascal

The roulette wheel contains numbers 1-36 but also contains 0 and 00 (in the U.S. version)

If you put $1 on the winning number, you win $36, seems like a break even game

However, they add two more slots… it’s the 0 and 00 slots that give them the advantage

So the probability of winning isn’t 1/36 it is 1/38

The way we calculate expected value for the roulette wheel is the following

[Probability of winning x Payout] + [Probability of losing x cost to play] = Expected Value

(1/38) ($36-$1) + (37/38) x ($0 - $1) = -.0593 which means the expected value is -.05 cents which means they make a nickel

So odds are against you

However, in 1873, Joseph Jagger, an engineer and mechanic, wondered how perfectly the roulette wheels worked

The roulette wheels work well IF they are perfectly balanced

He went to Monte Carlo, hired six assistants, one for each of the casino’s six roulette wheels

They observed the wheels writing down all the numbers that came up in the 12 hours the casino was open

Jagger analyzed the numbers after 6 days of recording results on the 6 wheels and found that 5 wheels provided no bias BUT on one wheel, some numbers appeared more often

Jagger, then headed into the casino to do some gambling on that wheel


After the 1 st day he had amassed $70,000 and then $300,000 after four days

On the 5 th day, Jagger began to lose, now he would lose on the other days but this was a consistent losing that corresponded to a normal roulette wheel

He noticed that the scratch on the wheel he noticed on the previous days was not there any more

He realized the casino had switched the wheel over night

He found the imbalanced wheel and started to win again

The casino finally started move the “frets” each night so the imbalance favored different numbers

He finally quit with winnings of $325,000
Virginia Lottery

In 1992, some investors from Australia, found a way to manipulate probability for their favor

They learned of a lottery jackpot in Virginia that had climbed to $27 million

The lottery involved picking 6 numbers from 1 to 44

They did some quick probability calculations using Pascal’s triangle and factorials to find out the number of number combinations

44! = 44 x 43 x 42 x 41 x 40 x 39 = 7,059,052 ways of choosing 6 numbers from 44
6! 38! 6 x 5 x 4 x 3 x 2 x 1

Thus, the jackpot exceeded the possible combinations

The Australian investors quickly found 2,500 small investors willing to put up $3,000 each and if the idea worked, the yield would be about $10,800 for each person

There were other potential problems, the possibility another person purchased a winning ticket and the biggest problem was the purchasing of more than 7,000,000 tickets!

In the end, they had purchased only 5 million of the 7 million tickets

After the winning ticket was announced, several days passed with no one claiming the prize but this was simply because it took the group that long to find out that they had actually won

Once again, Virginia tried to deny them the prize but in the end the investors won their money
Clown War

One of the major appeals to game theory is that many of the results are once again counter-intuitive to human reason.

Let’s take for example a problem given in a 1948 American Mathematical Monthly math periodical.

The situation here is we have three clowns who I have given the names Gonzo, Binky and Chuckles.

Let’s assume they are engaged in a game of darts where they alternate turns throwing darts at each other’s balloons. Each clown has one balloon and remains in the game as long as his balloon is not burst.
Each round the clowns draw lots to see who goes first giving the order of throwing to always be random
Let’s also assume the following skill percentages for each clown’s throwing abilities
Gonzo hits 80% of the time

Binky hits 60% of the time
Chuckles hits 40% of the time

If they all adopt the “obvious strategy” who would win? o We are tempted to think the best thrower would win but again the results are counterintuitive… let’s see o Because both Binky and Chuckles gang up on Gonzo, he is the LEAST likely to win o Let’s shake it up a bit

Gonzo and Binky start seeing that Chuckles has the best chances because no one is throwing at him in the first round so they agree to gang up on Chuckles until he is eliminated… what are the results now?

Gonzo, the best thrower, still is not the most likely to win because Chuckles will obviously aim for him.

So Gonzo threatens Chuckles and says that if he throws at him, he will retaliate so they make a truce for the first round… now both Gonzo and Chuckles aim at Binky and the probabilities change yet again

Binky then threatens Chuckles so that now both Gonzo and Binky have threatened him so surprising
Chuckles best line of action is to now “aim for the trees” and wait until the other two battle it out…

Game theory applies in many other situations, political elections, economics… it is also famous in what is known as the prisoner dilemma

Such dilemmas are of great interest to our students and really cause them to think critically about contextual details to probability
Stock Scam

An understanding of probability can help us detect when things are too good to be true to question things where others accept them at face value

Allow me to pose the following situation, let’s say you received an email from some random guy called stockprobilly@hotmail.com saying that you ought to buy IBM stock because it will increase in the next week.

What would you do? Obviously disregard it…

Then let’s say you received an email the second week with another prediction… you shouldn’t buy
Yahoo stock because it will go down in the next week…

Let’s say you get 6 emails in a row correctly predicting the outcome of the stock

He then proposes a deal. If you pay him $500, he will give you the next tip, fully refundable, if he is wrong.

I’m thinking, right now, you are 6 for 6 so ol’ Billy must be an incredibly savvy stock analyst

But let’s think of the situation, at first we were thinking scam but right now we can’t ignore the fact
 that he has correctly predicted this 6 times in a row.
Well, if we are a thinker we know this really is too good to be true… we might not be able to pinpoint the scam but we know it exists some where


Where is it? Let’s find out.

Stock Pro Billy is a smart guy, a mathematician but not very moral and a guy with way too much time on his hands.

He plans the scheme like so… o The first week he sends a total of 32,000 emails, 16,000 predicting that IBM goes down and
16,000 predicting it will go up o Well, 16,000 get the wrong prediction so these 16,000 are dropped from his email list (these
16,000 don’t know how fortunate they are) o The next week he sends 16,000 emails to the group that received the correct prediction and now he sends 8,000 the prediction that that Yahoo goes up and the other 8,000 get an email predicting it will go down. o Again, half the group might have been interested with 1 week of success but this second week puts him at 50-50 so Billy drops them from his list. o He still has 8,000 though that have received two correct predictions. o He keeps predicting and dropping the half that receive the wrong prediction. o After 6 weeks he will be left with 500 very unlucky victims who have seen 6 correct predictions and are ready for the trap. o He offers the deal and they think they are dealing with a pro and send in their $500. If they all pay, Stock Pro Billy is $250,000 richer
No Whammies!

Press your luck was a game show in the mid ‘80’s o The show was best known for the catch phrase “no whammies”…. “Big bucks, big bucks, no whammies, no whammies, stop!” o During the show, a big board with several squares would be randomly changing and the lighted square jumped around as well o The goal for the contestants was to press the buzzer and hope that they luckily selected a big money square and hopefully not a whammy square o If they landed on a whammy, they lost

Michael Larson was an ice cream truck driver, but basically unemployed, from Ohio o The ice cream business wasn’t so hot, watched Press your Luck every day o He asked a very good question to himself… was it truly random the sequence of the board o He taped the show, slowed it down and watched frame by frame and found out that there was a pattern (math – the science of patterns) o He memorized the patterns and practiced at home and realized he could, if he was patient, stop the board on the “big money” squares whenever he wanted

He went to Hollywood and was able to get on the show o In the process of the show, he had 47 “spins” and correctly hit the pattern on 41 of them and missed on 6 of them… at one point, he hit 30 rightly timed spins for “big money” o At the end the show, he had won $110,000, a record total for any game show episode at the time


The probability of this happening is simply too good to be true o If watching, you would be wondering how this could be happening, there was more to the story o CBS ended up trying to find a way in which they wouldn’t pay him but he technically had not broken any rules and they had to give him the money

Larson, sadly, went on to all sorts of corrupt schemes and ended up living on the run from the authorities

He passed away in Apopka, Florida in 1999
Millionaire Scandal

We all know the show who wants to be a millionaire

On the UK version, there was a particular episode where a contestant named Charles Ingram progressed through the game answering question after question right o But unlike other winning contestants, his thought process to correct answers was very bizarre o On repeated questions, Ingram appeared to have been headed down the wrong road to an incorrect answer but would inevitably change and get the right answer
For the £500,000 question, he was asked: "Baron Haussmann is best known for his planning of which city?
Rome, Paris, Berlin, Athens."
Ingram: "I think it is Berlin…. (says some other stuff) "I do not think it's Paris."
Cough.
Ingram: "I do not think it's Athens, I am sure it is not Rome.
"I would have thought it's Berlin but there's a chance it is Paris but I am not sure.
"Think, think, think! I know I have read this, I think it is Berlin, it could be Paris.
"I think it is Paris."
Cough.
[he goes on to get the question right]
The final question was: "A number one followed by 100 zeros is known by what name?"
A googol, a megatron, a gigabit or a nanomol.
Ingram: "I am not sure."
Tarrant: "Charles, you've not been sure since question number two."
Ingram: "The doubt is multiplied.
"I think it is nanomol but it could be a gigabit, but I am not sure.
"I do not think I can do this one.
"I do not think it is a megatron. I do not think I have heard of a googol."
Cough
Ingram: "Googol, googol, googol.
"By a process of elimination I have to think it's a googol but I do not know what a googol is.
"I do not think it's a gigabit, nanomol, and I do not think it's a megatron.
"I really do think it's a googol.
Tarrant: "But you think it's a nanomol, you have never heard of a googol."
Ingram: "It has to be a googol."
[this question was the 1 million pound question which he got correct]


True, improbable things occur but this sounds fishy

When producers of the show reviewed the tapes, they heard a faint but noticeable cough that was coming from the contestants surrounding the stage

One of the contestants waiting in the fastest finger circle was a game show junkie, like Charles
Ingram

When they amplified the coughs, they noticed that the coughs were coincidentally coming right after the correct answers while Ingram spoke aloud the different answers

So when he said “I do not think it’s Paris.” [cough]… he then changes his tone a bit to “… but there’s a chance it is Paris…”

“I do not think I have heard of a googol” [cough]… “I really do think it’s a googol”

Too good to be true


“It is likely that unlikely things should happen.” – Aristotle
“A tendency to drastically underestimate the frequency of coincidence is a prime characteristic of innumerates.” – John Allen Paulos
Stock Market Genius

Between the years 1979 and 1998, Leonard Koppett correctly predicted in 18 of 19 years whether the stock market would rise or fall
sounds like a pretty savvy investor
we would certainly listen to a guy with such a track record
what was his complex system of determining the results
Koppett, in reality was no investor at all, but rather a columnist for Sporting News
His system was based on the results of the Super Bowl o If a team from the (original) National Football League won, the stock would rise o If a team from the (original) American Football League won, the stock would go down
This ingenious system was successful 18 of 19 years
What a coincidence, what are the chances? Right?
True, when he begun his experiment the likelihood that he would do that was very unlikely.
But, think of it this way, let’s say 1 million people in 1979 all developed different random ideas on how to predict the outcome of the stock market, one person said it is dependent on whether or not the groundhog appears on February 2 nd , another says that if there is a full moon on Christmas eve, another that if the Orlando Magic have a winning season, the market goes up…. o If there are 1 million such ploys, the probability obviously increases dramatically and we should NOT be surprised to see such “coincidences”

Expecting the Improbable

Some of you also might be familiar with the German octopus, Paul the octopus, who correctly predicted the outcome of all 7 of Germany’s matches in the 2010 World Cup
we all watched in fascination
fun story, interesting
BUT, not necessarily that surprising!
Such schemes are being done all the time and obviously we aren’t hearing about Gilbert the
Goat who has correctly predicted the winner of 2 of the last 14 super bowls (fictitious story)

Was it probable that the Baltimore Orioles would have a 21 game losing streak at the start of the
1988 season? No!

Was it probable that a MLB team would have a 21 game losing streak at some point in the last 100
 years? Yes!
Similar comparisons can be made to a sports player who has a great game, made 6 three pointers in a row… this is referred to as the “hot hand fallacy”.
“Chance, too, which seems to rush along with slack reins, is bridled and governed by law.” Boethius
Buffon’s Needle

Buffon’s Needle is one of the oldest problems in the field of geometrical probability.

It asks the question that if, for example (there are different application), one were to have a lined sheet of paper with lines parallel and equal distance apart, what is the probability of randomly throwing a needle and having it cross one of the lines.
Two possibilities; a) it lands in between lines b) it lands where it is crossing the line

There are several different aspects to this problem, we will look at one of the easier.

Let’s assume the length of the needle is the same distance as the distance between the lines

By employing a little bit of integral Calculus, it can be found that
 p


1
/ 2

0


0
/ 2 l sin d
 d
 
2
 l d
(
 cos

)

0
/ 2 
2
 l d
Or simply 2/π… which means this funny little concept can be developed into an experiment to determine an approximation of Pi
Because
Probability = 2/ π
(Hits)/(Total Drops) = 2/ π
The value determine through integral calculus1
Probability = (Hits)/(Total Drops)
(Total Drops)/(Hits) = π /2 Reciprocal Property

2(Total Drops)/(Hits) = π Multiplication Property

What we need to do is perform an experiment and get some values to plug in for “total drops” and
“hits”

If you were to try, with your class, to estimate Pi, you could do the following. o Purchase some toothpicks that have a length of x. o On a poster board (or several), draw parallel lines that are x distance apart o Set out on the floor a piece of paper with parallel lines x cm apart. o Have each student do 50 tosses (or 100, depends on how accurate you desire to be) and have them record how many of their 50 tosses cross fall on top of a line o Therefore, let’s say you have 15 students in a class and each performs 50 drops for a total of
750 total drops. o Let’s assume that 476 of those drops hit the line. You would do the following estimation of
Pi:
2(750)/(476) = 3.1512606 (not bad!)* o *Be careful, don’t be disappointed if you get 3.5 or 2.9, remember, anomalies are likely to occur but if you do the exercise 20 straight years and your average is not close to Pi (you might be doing something wrong) o They also have online simulators that can illustrate buffon’s needle experiment

Here’s the point! An apparently random process generates order and produces a number very familiar to other areas of mathematics.
Driving Contest

In a recent year, the National Highway Traffic Safety Administration, calculated that the total amount of miles driven by about 200 million US drivers was 2.86 trillion miles.
that’s about 14,300 miles per driver

Let’s say that the country decided together to have a contest and attempt to hit the exact same total, 2.86 trillion miles in the next year
two methods could be employed to achieve this goal
Method 1 – the government institutes a rationing system assigning personal mileage targets for every US driver and then offer incentives if they can verify that this goal was reached
Method 2 – the authorities could do nothing and not make any effort at all but simply tell drivers to drive as much or as little as they please
Which method would come closer to our target?
Obviously method 1 is impossible to test but Method 2 was tested ironically, the very next year, no such effort was made and they totaled 2.8 trillion miles and more importantly about 14,400 miles per driver
The number of fatalities in year 1 was 42,815 versus 42,643 in year 2


What’s the point? We associate randomness with disorder and chaos, but the random behavior of millions of people proves very orderly

My goal with this seminar was to hopefully excite other teachers with the many exciting aspects of probability theory and encourage you all to bring this wonderful world to your classrooms

Even when we don’t use the words God and Jesus Christ we are preaching the gospel

If you can show a love for the order and patterns found in the realm of probability theory, you are showing a love for learning more about God’s creation

The Vision of this Seminar o Probability is very prevalent in our culture o Thinking through probability sharpens the minds of our students o Studying probability helps eliminate number and probability literacy pervasive in our culture o Unveil the order amongst the chaos o Cultivate a joy of mathematics
Mlodinow, Leonard. The Drunkard’s Walk: How Randomness Rules our Lives. New York: Pantheon
Books, 2008.
David, F.N. Games, Gods, and Gambling. New York: Dover Publications, 1998.
Paulos, John Allen. Innumeracy. New York: Hill and Wang, 2001.
Gardner, Martin. The Colossal Book of Mathematics. New York: W.W. Norton & Company, 2001.
The Joy of Thinking. The Teaching Company Video Series. 2008.