Transcript Leftoversx

Leftovers
The Gambler’s Fallacy
Independent Events
Two events A and B are independent if A does not raise the probability
of B and A does not lower the probability of B.
Pr(B/ A) =Pr(B)
Games of Chance
Games of chance, like dice-rolling, coin-flipping, roulette, etc. involve
random, independent events.
What side a coin lands on is independent of what happened on
previous tosses.
Tossing Coins
Let HHHHH abbreviate a run of 5 tosses of a coin, each of which lands
heads.
Pr(HHHHH) = 1/25 = 1/32
Imagine that you toss a fair coin 5 times and it lands heads each time.
That’s pretty rare!
Six Tosses
What is the probability that the next toss will land heads? You might
think it was even more unlikely, because:
Pr(HHHHHH) = 1/26 = 1/64
And yes, it is very unlikely for a coin to land heads six times in a row!
But…
But we are not interested in Pr(HHHHHH)! We are interested in
Pr(Heads on next toss/ HHHHH)– the probability the coin will land
heads assuming that it landed heads five times in the past.
Two Possibilities
Tosses
First Second Third Fourth
Possibility 1
H
H
H
H
Possibility 2
H
H
H
H
This is the past.
It cannot change.
Fifth
H
H
Sixth
H
T
Two Possibilities
Tosses
First Second Third Fourth
Possibility 1
H
H
H
H
Possibility 2
H
H
H
H
Fifth
H
H
Sixth
H
T
One in two chance of
heads next flip.
The Gambler’s Fallacy
The gambler’s fallacy is when someone assumes that if something
happens more often in the past, it will happen less often in the future.
Monte Carlo
On 18 August 1913, the ball on
one of the roulette wheels at the
Monte Carlo Casino in Monaco fell
black 26 times in a row.
Improbable!
In roulette, the ball has a 50-50
chance of landing red or black. So
the probability of landing black 26
times in a row is:
1/226 = 1/67108864
Improbable!
But in fact, the probability of any
sequence of reds and blacks is:
1/226 = 1/67108864
And the probability of landing red
(or black) after any sequence is
always 1/2.
Losing Money
On that night in 1913, gamblers
bet millions of French francs
against black.
Each time the wheel landed black,
they became more and more
certain it would land red the next
time, and bet increasingly large
sums… which they lost.
Children
People commit the gambler’s fallacy frequently. A lot of people think
that if you have three daughters, your next child will be a boy.
But the probability of having a boy is always 50%. It’s independent of
the children you previously had.
Necessary and Sufficient
Conditions
B
Every A is a B
A
Animal
Every dog is an animal
Dog
Necessary and Sufficient Conditions
Every dog is an animal:
Being an animal is a necessary condition for being a dog.
Being a dog is a sufficient condition for being an animal.
Necessary Conditions
A necessary condition for X is any condition you must have if you are X.
If you are a dog you must be an animal. So being an animal is a
necessary condition for being a dog.
If you are a triangle you must have three sides. So having three sides is
a necessary condition for being a triangle.
Not Necessary
To show that some condition is not a necessary condition for X, you
show that you can be X without having that condition.
Example: Dr. Asay once asked students to give a necessary condition for
life. One student wrote “pork.” (I assume he or she meant “eating
pork.”)
But you can be alive without eating pork. There are living vegetarians!
Sufficient Conditions
A sufficient condition for X is any condition whose presence is
guaranteed by X.
If you are a dog, you are guaranteed to be an animal. So being a dog is
a sufficient condition for being an animal.
If you are a square, you are guaranteed to have four sides. So being a
square is a sufficient condition for having four sides.
Solving a Problem
Which of the following is true?
a. Being a dog is necessary and sufficient for being an animal.
b. Being a dog is necessary but not sufficient for being an animal.
c. Being a dog is not necessary, but it is sufficient for being an animal.
d. Being a dog is not necessary and not sufficient for being an animal.
Solving a Problem
Which of the following is true?
a. Being a dog is necessary and sufficient for being an animal.
Every animal is a dog and every dog is an animal.
b. Being a dog is necessary but not sufficient for being an animal.
Every animal is a dog but not every dog is an animal.
c. Being a dog is not necessary, but it is sufficient for being an animal.
Not every animal is a dog, but every dog is an animal.
d. Being a dog is not necessary and not sufficient for being an animal.
Not every animal is a dog and not every dog is an animal.
Solving a Problem
Which of the following is true?
a. Being a dog is necessary and sufficient for being an animal.
Every animal is a dog and every dog is an animal.
b. Being a dog is necessary but not sufficient for being an animal.
Every animal is a dog but not every dog is an animal.
c. Being a dog is not necessary, but it is sufficient for being an animal.
Not every animal is a dog, but every dog is an animal.
d. Being a dog is not necessary and not sufficient for being an animal.
Not every animal is a dog and not every dog is an animal.
Remember
X is a necessary condition for Y = Every Y is X
X is a necessary condition for Y = If you are Y, then you are X
You cannot be Y without being X
X is a sufficient condition for Y = Every X is Y
X is a sufficient condition for Y = If you are X, then you are Y
X’s must be Y’s
Definitions
Types of Definitions
If you open up a dictionary, you
find a bunch of definitions.
But not all definitions are trying to
do the same thing. There are
different types of definitions.
Example
Uncle: a male brother of one’s father.
This definition gives an expression “male brother of one’s father” that
means the same thing as the expression “uncle.”
Being a male brother of my father is a necessary and sufficient
condition for being my uncle.
Example
Uncle: a male brother of one’s father.
We call this definition a “descriptive” definition because it (accurately)
describes the meaning of the word it defines.
Example 2
Red: the color of a ripe tomato.
This definition is correct in the
following sense: everything that is
red is the same color as a ripe
tomato and everything that is the
same color as a ripe tomato is red.
Example 2
Red: the color of a ripe tomato.
But this is not the meaning of the
word ‘red.’ If tomatoes started
ripening blue, that would not
mean that water was red and the
sky was red and blueberries were
red…
Example 2
Red: the color of a ripe tomato.
We call definitions like this
“extensional” definitions. They
allow you to pick out everything
you can truly describe with the
word (for example, ‘red’) without
giving you the meaning of the
word.
Stipulative Definition
A stipulative definition does not report the meanings of words as they
are actually used.
Instead it introduces a new meaning, either for an already existing
word, or for a new, made-up word.
Spotting Stipulative Definitions
There are many ways people have of indicating that they are stipulating
a meaning:
• “Let us define X as…”
• “In what follows, we will understand X to mean…”
• “Suppose we say that a thing is X when…”
Purposes of Stipulative Definitions
Sometimes there is not a word for what you want to talk about, so you
have to invent one.
Science and math are full of words with stipulative definitions.
(“prime,” “composite,” “whole,” “continuous”…)
Precising Definitions
Precising definitions are a combination of descriptive and stipulative
definitions. This is when you take the normal meaning of a word, and
stipulate some aspects of its meaning.
“Criminals cannot apply for this job.”
“Students get a set lunch discount.”
Using Precising Definitions
Precising definitions can often be used to resolve certain arguments.
“If by ‘marriage’ you mean that gays can have a religious ceremony, live
together, and call each other ‘spouse’ then yes, gays can get married in
our country; but if you mean by ‘marriage’ a recognition by the
government and society of the validity of such a relationship, then they
cannot.”
Summary
Descriptive definition: gives the meaning of a word
Extensional definition: does not give the meaning; categorizes the
things that the word applies to
Stipulative definition: introduces a new meaning for a word
Precising definition: takes the old meaning of a word and changes it in
some way
For the Sake of Argument
Assumptions for the Sake of Argument
Sometimes we make assumptions for the sake of argument.
We either know that the assumption is false or we don’t know whether
it is true or false.
But in some cases it is useful to make false or unknown assumptions…
Conditional Proof
Suppose you want to prove a conditional (“if-then”) statement.
For example, suppose you want to show that if the accuser is telling the
truth, then the accused should go to jail.
The Accusation
Michael kicked me.
Assuming for the Sake of Argument
First, you would assume for the sake of argument that the accuser is
telling the truth. Assume that Michael did in fact kick the puppy.
(Even though of course he’s innocent.)
Conditional Proof
Then you would use that assumption to show that Michael belonged in
jail.
You would argue that since kicking puppies violates article 2, section 6,
paragraph 3 of the criminal code, Michael belongs in jail.
Conditional Proof
Finally, you would stop assuming that Michael did actually kick the
puppy and conclude:
If the accuser is telling the truth, then Michael belongs in jail.
Assume the Antecedent
If the accuser is telling the truth, then Michael belongs in jail.
Assume this
Prove the Consequent
If the accuser is telling the truth, then Michael belongs in jail.
Prove this
Conclusion
If the accuser is telling the truth, then Michael belongs in jail.
Conclude this
(No longer assume
the antecedent)
Reductio ad Absurdum
Another type of argument that makes assumptions we don’t believe is
reductio ad absurdum (reduction to absurdity or proof by
contradiction).
This is a common type of argument form in mathematics.
Example
Here’s a proof that there is no smallest rational number (number that
can be expressed as a fraction.
Assume: There is a smallest rational number, N.
Use mathematical reasoning: If N is rational, N/2 is rational
Use mathematical reasoning: N/2 < N
Conclude: N is not the smallest rational number.
The assumption is false because it leads to an absurdity!
Reductio
Step 1: Assume the opposite of what you’re trying to prove.
Step 2: Prove something absurd by using that assumption.
Step 3: Conclude the opposite of your assumption.
Real Life Example
According to the defendant, he was at the library at the time of the
crime.
However, the crime occurred at 22:30, and the library is not open at
that time.
Therefore, the defendant was not at the library!