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 Fifth Sixth

Possibility 1 H H H H H HPossibility 2 H H H H H T

This is the past. It cannot change.

Two Possibilities

Tosses First Second Third

Fourth Fifth Sixth

Possibility 1 H H H H H HPossibility 2 H H H H 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 MoneyOn 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

Every A is a B

A

B

Every dog is an animal

Dog

Animal

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 NecessaryTo 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 ConditionsA 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 ProblemWhich 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 ProblemWhich 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 ProblemWhich 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 XX 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 YX 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 DefinitionsPrecising 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 wordExtensional definition: does not give the meaning; categorizes the things that the word applies toStipulative definition: introduces a new meaning for a wordPrecising 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.

ExampleHere’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 rationalUse mathematical reasoning: N/2 < NConclude: 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!