## Philosophy Club Tryouts

Fellow Badasses,

Let it be known that the Philosophy Club is having an organizational meeting on Wednesday 11/9 from 7:30-9:30 in room 221 Strickland. Students who are interested in learning more about the club should contact one of our majors, Kyle Hendricks (klh6x2@mail.missouri.edu).

Crazy things happen in philosophy club. Great things. Terrible things. But great things.

## Schedule before Exam 3

Nov. 1 (Tuesday): Chapter 13, Vagueness.

Nov. 3 (Thursday): Chapter 14: Ambiguity.

Nov. 8 (Tuesday): Chapter 15: Relevance and Vacuity

Nov. 10 (Thursday): Chapter 16: Vacuity

Nov. 15 (Tuesday): Review

Nov. 17 (Thursday): Exam 3

## Lecture 10/20: More on Chances

**Probability Theory (Chapter Ten)**

Continued from last time:

**E. Availability Heuristic**

Number of 7-letter words ending in -ing vs. number of words ending in _n_. Because we can think of more words ending in -ing than we can (non -ing) words ending in _n_, we think the former will be more numerous than the latter. This is wrong, however, because all -ing words are _n_ words, so there will be at least as many of the latter.

Another example: who has a better batting average, NY Yankees or Boston Redsox? Many will think of the superstars and forget that the whole team contributes to the overall batting average of the team. The less famous players are not “available” to you, in the sense that you cannot think of them off the top of your head.

**Rules of Probability**

We write the probability of h, for ‘hypothesis’, as Pr(h). The Pr(h) = the number of outcomes favorable to h over the number of total outcomes; favorable/total.

1. **Negation**: Pr(~h) = 1 – Pr(h). The probability that a hypothesis is false is equal to 1 minus the probability that h is true. If the Pr(h) = .4, then the Pr(~h) = 1 – .4 = .6.

2. **Conjunction with Independence**: Pr(h1 & h2) = Pr(h1) x Pr(h2). Given two independent events, the probability of both occurring is figured by conjunction with independence. Independence refers to whether the outcome of one event gives you any information about the outcome of the other event. For example, if you draw a card from a normal deck, put it back and shuffle it, then the outcome of the next draw is independent of the first; both outcomes have a probability of 1/13. However, if you draw a card, keep it out, and draw a second card, then the information from the first event tells you something about the outcome from the second event. The probability of drawing two kings, by drawing a card, putting it back and shuffling, and drawing another, is: Pr (h1 & h2) = Pr(h1) x Pr(h2) = 1/13 x 1/13 = 1/169.

2G.** Conjunction in General:** To extend the rule to cover events that are not independent, we need the idea of **Conditional Probability**. This is the probability that something will happen, **given** that some other thing happen, i.e., dependent on something else happening. If we want the probability of h2, given that h1 happened, we write Pr(h2|h1). For example, we may want to know the probability that we draw a king (h2), given that we just drew the king of diamonds (h1). Conditional probability is figured out by considering the outcomes where both h1 and h2 are true, divided by the total h1 outcomes. The rule for Conjunction in General is: Pr(h1 & h2) = Pr(h1) x Pr(h2|h1). The probability that you draw two kings in a row without replacing the first is 4/52 x 3/51 = 1/221. The probability that you draw a king, given that you’ve just drawn a king, is the conditional probability. It is 3/51, because there are 3 favorable outcomes when you’ve already drawn a king, over 51 total outcomes where you’ve already drawn a king. Conjunction with independence is a special case of conjunction in general.

3. **Disjunction with Exclusivity:** Pr(h1 or h2) = Pr(h1) + Pr(h2). The probability that one of two mutually exclusive events is the sum of the probability of each. The probability you roll a 5 or an 8 (Jumanji reference!) is Pr(roll a 5) + Pr(roll an 8 ) = 4/36 + 5/36 = 9/36 = 1/4. Pretty decent chances of getting out the jungle.

3G. **Disjunction in General:** Of course, not all either/or statements are exclusive. Many are inclusive, meaning that it is possible for both to occur. Thus, we need a general formula for figuring out disjunctive probabilities. It is Pr(h1 or h2) = Pr(h1) + Pr(h2) – Pr(h1 & h2). Suppose half the class are male, and half female, and that half are over 19, and half are under or equal to 19. If we want to know the chances that someone is either female or over 19, we figure the Pr(h1) = 2/4, plus the Pr(h2)= 2/4, minus the Pr(h1 & h2) = 1/4. 2/4 + 2/4 – 1/4 = 3/4. So, the probability that someone is either female or over 19 is 3/4. Disjunction with exclusivity is a special case of disjunction in general.

4. **At Least**: The probability that an event will occur at least once in a series of *n* independent trials, where *n* is the number of trials, is 1 – Pr(~h)raised to the *n*th power. What are the chances of tossing heads at least once in 8 independent flips of a fair coin? Restate the question so that rules 1 and 2 can be used. First, what are the chances that we don’t flip at least one heads? That is 1 – Pr(flip at least one heads). This is the same as saying the probability of flipping 8 tails in a row. That’s Pr(tails) x Pr(tails) x Pr(tails) x Pr(tails) x Pr(tails) x Pr(tails) x Pr(tails) x Pr(tails) = Pr(tails)to the 8th power = 1/256. So, 1 – Pr(flip at least one heads) = 1/256. Those are the chances we *don’t* flip at least one heads. So, Pr(flip at least one heads) = 255/256. Pretty good chances! So, to calculate ‘At Least x’, you start by asking the chances that you DON’T get at least x: this is 1 – Pr(at least x). This is the same as asking the chances that the alternatives to x happen *n* times in a row, which is just an application of rule 2 or 2G above, depending on whether it is independent or not. Then, remember to reconvert it to the original question by figuring 1 minus whatever you got.

## Lecture 10/18: Taking Chances

Notes on taking chances

**Probability Theory (Chapter Ten)**

I. Gambler’s fallacy and the law of large numbers

A. Examples

B. Law of Large numbers: The difference between the observed value of a sample and its true value will diminish as the number of observations in the sample increases.

Applications

O: after a billion flips of a coin, we counted 48% heads, 52% tails.

H1: Fair coin

H2: coin is weighted towards tails

Which hypothesis is predicted by the law of large numbers?

answer: H2, due to the Law of Large Numbers

O: after ten flips of a coin, we counted 6 heads and 4 tails

H1: Fair coin

H2: Coin is weighted towards heads?

Which hypothesis is predicted by the law of large numbers?

answer: predictively equivalent (H1 = H2)

Random sequences:

A. 1, 1, 1, 1, 1, 1, 2, 1, 1, 2

B. 1, 2, 1, 2, 2, 1, 2, 1, 1, 2

C. 1, 2, 2, 2, 2, 1, 2, 1, 1, 2

D. 1, 2, 2, 2, 1, 2, 1, 2, 2, 1

Which one was generated by a “randomizer” ? A

http://www.randomizer.org—form.htm

Question: Suppose you flip a fair coin three heads in a row. What is the

probability that a head will come up a fourth time?

answer: 1/2

C. Misapplication of law of large numbers

Example 1 and 2:

law of large numbers does not support the idea that a gambler will experience runs of good luck after a run of bad luck. For coins and casino machines the probability of any outcome is independent of the number of trails you have experienced. All bets are off if the trials are dependent rather than fair (but then no one would play at such a casino where the outcomes are rigged).

D. Examples outside of gambling

– Hot streaks in basketball: give the ball to the buy who has made a bunch of shots in a row. But, statistically, hitting three or shots in a row is statistically insignificant.

– “market beaters” in fund managing: you swap out of your underperforming funds and into the hot fund. But, given that the market is pretty efficient, past performance is not a good guide to future performance (there will be streaks for any fund over long enough period of time).

**II. Common judgements and their fallacious foundations**

A. Confirmation bias: you are convinced beforehand that a stock picker or basketball player can “get hot” (due to media attention or your own feelings about the person). So, you ignore the fact that streaks are likely in the short term (given law of large numbers).

B. Over optimism

How often does a college basketball team that is trailing at halftime come back to win?

answer: (less than 20%) (people typically guess 30%-60%)

data: 3300 games in Nov-Jan.

Why are we often wrong?

We are optimistic and media gives most attention to comeback victories.

C. Irrationality due to desire to win

Suppose 50% chance of scoring on a two-point shot. 33% for a three-point shot. A team is down by two points and it has time for one last shot. What play should the coach call?

answer: if the team makes the two-point shot, it still has to play overtime, where its chances of winning are 50%. Have to win on two 50% gambles = 25%

overall. So, should go for 3-points.

Apply this to stocks: many investors shy away from stocks because of the potential for short-term sting (like the sting of losing from a 3-point shot at the

buzzer). But, in the long run stocks are best investment (over its history).

D. Representative heuristic

question #1 on Tversky teasers.

People tend to say that Hand #2 is much more unlikely than Hand #1. But, each is equally likely in a fair game.

Representative heuristic: Hand #2 is more unimpressive so it is more likely to represent an ordinary hand.

question #2 on “teasers”

89% of students said that it is more likely that Linda was both a bank teller and a feminist than that she was simply a bank teller.

can’t be true: the probability of two things being true can never be higher than the probability that just one of them is true (one is true if both are).

## Stochasticity AKA Randomness

Everyone will listen to http://www.radiolab.org/people/jay-koehler/# and achieve enlightenment… OR ELSE!!

What do you think?

Also, here is a brief article by Jonathan Clements on interesting stuff that is good to know:

“IF YOU WANT to improve your investing, settle into an armchair, grab the remote — and spend the next few weeks watching the NCAA basketball tournaments.

True, this might not endear you to your family. But the fact is, there are intriguing parallels between the foibles of basketball and the behavioral mistakes investors make.

As you cruise the channels, what should you look for? Academic and financial experts offer up these three basketball-inspired investment insights.

— Keeping your cool. Both the men’s and women’s NCAA tournaments begin this week. If you think stock jockeys are obsessed with finding the next hot thing, just listen to the basketball commentators.You will probably hear the television announcers declare that one or two of the players have the “hot hand” because they have scored on, say, their last three shots. The implication: Their teammates should feed them the ball, because there’s a good chance they will keep knocking down the jump shots.

Academics, however, would beg to disagree. A study that appeared in Cognitive Psychology in 1985 looked at the shooting record of the Philadelphia 76ers during the 1980-81 season, a squad that included the great Julius Erving. The study found that, contrary to popular belief, the probability that the players would score on their next shot was, on average, slightly lower following a successful shot.But what about those unusual hot streaks? Statistically, hitting three or four shots in a row — or beating the market in consecutive years — just isn’t that unusual. Indeed, if you and a bunch of friends each flipped a coin 20 times, half of you would likely get four heads in a row.

“Fund managers can look like they’re hot or like they’re a market beater,” says Thomas Gilovich, co-author of the “hot hand” study and a psychology professor at Cornell University. “But you swap out of your underperforming fund and into the hot fund at your peril. Given that the market is pretty efficient, past performance just isn’t a good guide.”Why do people reach grand conclusions based on skimpy data? Part of the blame lies with so-called confirmation bias. If you are convinced you’re a great stock picker or that basketball players can “get hot,” you will likely find the necessary proof.

“The brain looks for patterns,” says Meir Statman, a finance professor at Santa Clara University in California. “And once you decide there is a pattern, you will look for confirming evidence and you will dismiss contradictory evidence as a fluke.”— Expecting less. While it’s hard to say definitively that some fund managers are superior to others, some basketball teams clearly are more skillful. Yet fans of weaker teams are forever hopeful.

How often does a college basketball team that’s trailing at halftime come back to win? Allan Roth, a financial planner with Wealth Logic in Colorado Springs, Colo., often puts this question to audiences. He says people typically guess that between 30% and 60% of teams make a comeback.

In fact, Mr. Roth looked at over 3,300 college games played in November, December and January and found that, among teams trailing at the half, less than 20% came back to win. Why do folks think the number is so much higher? Mr. Roth figures there are two reasons.First, we tend to be overly optimistic. “It’s America,” Mr. Roth says. “We believe in the underdog — and we believe in the small investor.” Even though studies suggest that most investors lag far behind the market, we like to think we can beat the odds and come out on top — which helps explain why market-tracking index funds still aren’t that popular.

Second, comeback victories tend to get the most media attention, so they stick in our minds. “It’s the same thing with hot mutual funds and hot money managers,” Mr. Roth says. “Because investors only hear about the winners, they think it’s easy to beat the market.”

— Playing the odds. Investors hate the idea of losing. So, too, do basketball coaches — and it can lead both groups to be a little irrational.

Suppose a team is down by two points and it has time for one last shot. What play should the coach call? Let’s say there’s a 50% chance of scoring on a two-point shot and pushing the game into overtime, but only a 33% chance of making a three-point shot and getting the immediate win.

Nonetheless, the three-point shot is the rational choice. The reason: If the team makes the two-point shot, it still has to play overtime, where its chances of winning are 50%. In other words, by opting for the two-point shot, the team is looking at having to win on two 50% gambles, which means its overall odds of winning are just 25%.Yet coaches usually go for the two, notes Richard Thaler, an economics professor at the University of Chicago. Chalk it up to our aversion to regret. If the coach goes for the three and misses, not only will the team suffer an immediate stinging loss, but also critics will vilify the coach as “greedy” and “reckless.”

Similarly, investors are often too worried about looking foolish in the short term. Stocks, like the three-point shot at the buzzer, may be the best bet. But many investors shy away from stocks, because they worry about stinging short-term losses and the pangs of regret that accompany them.

“If you believe there’s a premium to owning stocks, you’re crazy not to own them if you’re a long-term investor,” Prof. Thaler argues. “You shouldn’t be so bothered by day-to-day or month-to-month volatility.”

## Lecture 10/11: IBE

We looked at a non-deductive argument technique called “inference to the best explanation” (IBE). Gregor Mendel’s famous experiments on peas are great examples of IBE.

Gregor Mendel (19th century)

Discoverer of GENES although, he never saw any?

Step 1 the experiment:

pea plants (what I’m about to say isn’t necessarily accurate, but captures the basic ideas.)

————-

Pea Variations:

purple vs. white petals.

smooth vs. wrinkled peas

tall vs. short stems

To begin, Mendel identifies a particular trait on which to experiment. He breeds each purple plant with a white plant, and gets:

generation 1: (purple vs. white petals)

all purple petals.

Thus, because he has a large, unbiased sample size, he concludes that every time you breed a purple pea plant with a white pea plant, you get a purple pea plant. Next, he breeds the members of generation 1 with each other, and gets:

generation 2: selfed the purple plants.

3:1 purple to white petals.

Time to get your science on! Mendel is freaking out, wondering, “How did the white petals return? Are they ghosts?” (no citation available). From all purple petal (people eaters) in generation 1, a ratio of 3:1 purple to white resulted. Again, due to his large unbiased sample size, he concludes the general claim: Every time you mix (what we now call ‘heterozygous’) pea plants with other heterozygous pea plants, you get a ratio of 3:1 on the traits.

Step 2: asks “why do I get 3:1 ratios?”

hypothesis (H1): suppose transmission ranges over particles (suppose there are genes) and they obey laws (L).

hypothesis (H2): traits in offspring are the result of a blending process of the parents’ traits.

[which hypothesis best explains what I just saw? i.e. 3:1 ratios]?

What sort of inference did Mendel do?

Deduction (nope, not deduction).

Ratio of purple to white is 3:1

—–

Therefore, there are genes that obey L.

that is akin to arguing as follows:

O: lock is broken, lots of valuable in your room are missing.

H: I was robbed! [invalid]

Deduction, take 2:

(1) If there are genes and they obey L, then I

would expect to see 3:1 ratios.

(2) I see 3:1 ratios.

(3) Therefore, there are genes and they obey

L.

if p then q

q

therefore, p.

——–

Bahhhh, Invalid.

Deduction, take 3

Modus tollens

if p then q

~q

Therefore, ~p

(1) If there are genes and they obey L, then I

would expect to see 3:1 ratios.

(2) I do not see 3:1 ratios.

(3) Therefore, there are no genes that obey L.

Whoa, that can’t be right. Mendel’s conclusion was that there ARE genes that obey L.

Is it induction?

While maybe the discovery of 3:1 ratios in crosses of pea plants is an example of induction, the discovery of GENES is not based on induction. Why? Because he didn’t observe any.

“postulating unobservables”: proposing the existence of something that has not been observed, or cannot be observed directly.

• deduction can’t do that.

• induction can’t do that.

• inference to the best explanation **can**.

See how inference to the best explanation

(IBE) works:

Step 1, mundane example.

O: lock is broken, lots of valuable in your room are missing.

H: I was robbed!

If H is true then O is likely to be true.

Strength of an inference to the best

explanation is how likely the observation is

given the truth of the hypothesis.

H1: I was robbed!

H2: the cops raided your place!

H3: your friend played a joke on you!

If H2 were true, then is O likely?

If H3 were true, then is O likely?

If H1 were true, then is O likely?

H1 is the best explanation for O.

Step 2: Mendel revisited.

O: 3:1 ratios of dominant to recessive features (purple vs. white petals, tall vs. short

plants, etc.).

H1: there are genes and they obey L

H2: Blending theory (like paint).

If H1 were true, is O is likely?

If H2 were true, is O likely?

H1 is a better inference than H2 given O.

So, Mendel is using the IBE, and IBE does some cool stuff, which we’ll learn about in the next lecture post.

## Battling Bad Science

Sorry about the previous link. Try this: http://www.ted.com/talks/ben_goldacre_battling_bad_science.html

What do you guys think about this?