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 nth 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.
Notes on taking chances
Probability Theory (Chapter Ten)
I. Gambler’s fallacy and the law of large numbers
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.
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)
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
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?
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).
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.)
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
if p then q
Deduction, take 3
if p then q
(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
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
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.
Although there were plenty of versions of this style of argument, William Paley’s (1743–1805) was and still is very influential. Darwin is said to have brought his copy of Natural Theology—where the passage comes from—along with him on his five-year voyage on the Beagle.
Unlike St. Anselm’s and St. Aquinas’s deductive arguments, this one is nondeductive. Whether it is an argument by induction, analogy, or inference to the best explanation is slightly controversial among philosophers. In what follows I’ll impose my own views on this—that Paley’s argument is indeed an inference to the best explanation. I’ll make the case in a rather lengthy discussion (I have adapted the discussion from a paper I wrote: Ariew, A. “Teleology.” In M. Ruse and D. Hull’s Cambridge Companion to Philosophy of Biology. Cambridge University Press. 2008.)
Paley’s argument starts with an analogy between living organisms and human artifacts. Briefly, if you come across a watch and inquire as to its existence, you would not take seriously the conclusion that watches are the product of natural forces. It is highly improbable that natural forces would randomly coalesce matter into a watch.
The possible existence of a designer who can manipulate the parts for his or her own purpose makes the existence of watches much more plausible. Paley’s conclusion is that the existence of a designer best explains watches and living organisms.
As I said above, in my view, Paley’s is an instance of an inference to the best explanation. Paley’s inference to the existence of watchmakers (and later, his argument for the existence of a divine creator) works exactly in this way. The existence of watchmakers is supported by the existence of watches because watchmakers best explain how such complex things could come to exist.
An interesting feature of inference to the best explanation arguments is that with them, we can infer the existence of unobservable phenomena. As Paley put it, already having inferred the existence of a watchmaker from inspection of a watch found on a dirt path:
Nor would it, I apprehend, weaken the conclusion, that we had never seen a watch made—that we had never known an artist capable of making one—that we were altogether incapable of executing such a piece of workmanship ourselves, or of understanding in what manner it was performed.
The strength of the inference to a watchmaker depends not upon our witnessing watchmakers making watches but in the relative likelihood that watches would exist if skilled watchmakers were to exist. Compare that to the likelihood of the materialist hypothesis that watches would exist only if the random action of natural forces were to exist. Natural processes alone are unlikely to make a watch. A creator with forethought more likely will.
The feature of inferring the existence of unobservable causes distinguishes inferences to the best explanation from garden-variety inductive arguments. Let’s see why. The strength of an inductive argument depends on the size and bias of the sample. Yet, as Paley suggests in the quote above, we may never have seen watchmakers make watches. If so, our sample size is zero—likewise for the sample of times in which any of us has seen God creating living things. If teleological arguments are inductively based, then they would be nonstarters. It’s a good thing for Paley that his inference is not inductive. In an inference to the best explanation, the issue is not the properties of a sample but rather what could explain an observed phenomenon.
Next, Paley considers what would happen if we found a self-replicating watch. The passage is a lovely early example of science fiction:
Suppose, in the next place, that the person who found the watch should after some time discover that, in addition to all the properties which he had hitherto observed in it, it possessed the unexpected property of producing in the course of its movement another watch like itself—the thing is conceivable; that it contained within it a mechanism, a system of parts—a mold, for instance, or a complex adjustment of lathes, files, and other tools—evidently and separately calculated for this purpose; let us inquire what effect ought such a discovery to have upon his former conclusion.
In addition to serving the function of telling time, this watch has a further extraordinary feature: it produces well-functioning offspring. The discovery of a self-replicating watch affects the former conclusion—the existence of watchmakers—in several important ways. First, it further illustrates that the strength of Paley’s inference does not depend on ever having seen any watchmakers make watches. As I argued above, this feature distinguishes inference to the best explanation arguments from inductive ones because it does not depend on sampling from a population of events.
Second, and more important, the discovery of the self-replicating watch strengthens the inference to the existence of a designer at the same time that it weakens the inference to the hypothesis that the item is the product of natural forces alone. As Paley puts it, “If that construction without this property, or, which is the same thing, before this property had been noticed, proved intention and art to have been employed about it, still more strong would the proof appear when he came to the knowledge of this further property, the crown and perfection of all the rest.” The probability of natural forces randomly producing a watch is very small, but the probability of natural forces randomly producing something as extraordinary and exquisite as a self-replicating watch is even smaller.
The general lesson is this: the more complex the parts, the stronger the evidence of a designer. It and the next feature play a large role in Paley’s ultimate inference, the existence of a God.
The third effect that the self-replicating watch example has on the former conclusion is, in Paley’s words, to “increase his admiration of the contrivance, and his conviction of the consummate skill of the contriver.” However complex watches are, most of us with average intelligence and skills could imagine learning, after extensive training, how to create watches. But to have the skill of a self-replicating watchmaker would be extraordinary or even supernatural. The general lesson here is that the more complex the design, the more intelligent or skillful the designer.
Once the second and third features of the new inference to the best explanation from self-replicating watches are in place, all that Paley has to convince us of is that living tissues, organs, organisms, and ecosystems are much more complex than self-replicating watches and that their parts are much more attuned to the functions that they serve. In a sense, that is the intention of the bulk of Paley’s book, Natural Theology, from which his famous argument for the existence of an intelligent designer is a relatively small section. The later chapters are more or less a zoological textbook, detailing the wonder of natural adaptations. The most famous passages are found in the section that describes the anatomy and function of the eye. For instance, he expresses amazement at how the anatomy of eyes from animals living in distinct environments differs according to the laws of transmission and refraction of rays of light. I’ll let Paley express the point himself (this is from later on in Paley’s book):
For instance, these laws require, in order to produce the same effect, that rays of light in passing from water into the eye should be refracted by a more convex surface than when it passes out of air into the eye. Accordingly, we find that the eye of a fish, in that part of it called the crystalline lens, is much rounder than the eye of terrestrial animals. What plainer manifestation of design can there be than this difference? What could a mathematical instrument maker have done more to show his knowledge of his principle, his application of that knowledge, his suiting of his means to his end—I will not say to display the compass or excellence of his skill and art, for in these all comparison is indecorous, but to testify counsel, choice, consideration, purpose?
Passages like this remind me of the many nature programs found on the Discovery channel (http://dsc.discovery.com/convergence/planet-earth/planet-earth.html) or PBS (http://www.pbs.org/lifeofbirds/) that showcase nature’s great adaptations. One would think, given the shows’ sponsorships, that the purpose of these shows is to educate the public about the wonders of a biological world that emerges from natural selection. But ironically, as passages like the one above from Paley remind us, there’s no reason to think that the very same phenomena featured on these nature programs could not equally serve as grist for the Creationists’ mill.
Does Paley’s argument succeed in proving that living organisms are created by God? Among his contemporaries, he provided a powerful argument against a materialist who seems to have little to account for natural adaptation. Yet, the success of inference to the best explanation arguments depend on the relative success of the given hypotheses, and, as Darwin would show, there are other hypotheses to consider besides the random action of matter and cause and an intelligent designer. Therein lies a formal limitation of inferences to the best explanation. The strength of an inference to the best explanation is only as good as the proffered hypotheses. For any given set of hypotheses, the interlocutor has always the option to remain agnostic as to the cause of the phenomenon in question. To suggest otherwise—for instance, to argue that because God is a better explanation than matter and cause, God must exist—is to commit “the only game in town” fallacy. As for proving the existence of an omnipotent and omniscient God, Paley’s commentators and critics often pointed out that while Paley’s inference might have strongly suggested the existence of a supremely intelligent designer (a supernatural designer), it stops short of proving that the designer is God.
This lecture focused on the following two main points:
1. Distinguish between inductive and deductive arguments.
2. Evaluate the strength of inferences from generalization.
Read Along: Understanding Arguments, Chapter 8 (pages 215–230)
The standard of validity is powerful because, as we’ve discussed in earlier lessons, valid arguments guarantee the truth of the conclusion if you provide true premises. As we have seen, it is easy to get muddled by invalid arguments or arguments that sometimes contain true conclusions and sometimes not, regardless of the truth of the premises. The truth of a conclusion of an invalid argument is more or less accidental, since invalid arguments do not come with a guarantee. Valid arguments provide that guarantee.
Yet, there is something about valid arguments that limit their applicability. Vaguely put, in a deductively valid argument, the conclusion cannot say anything that was not already contained in the premises. Let me explain.
First, you get a sense of what I mean by the phrase in italics by considering a valid argument form. Here’s one:
All As are Bs.
All Bs are Cs.
Therefore, all As are Cs.
Notice that the conclusion is a rearrangement, as it were, of the variables in the premises (not a haphazard rearrangement but one nonetheless). So, there is a sense in which we don’t learn anything new in the conclusion:
the information in the conclusion is already contained in the premises (just rearranged). The premises and the conclusion are all about As, Bs, and Cs. Now, what I’m saying is still vague because there is a sense in which you do learn something new in the conclusion of a valid argument. You learn new ways to understand the relationship between the variables. But, in another sense, valid arguments don’t tell us about anything outside of the content of the variables that are mentioned in the premises. Here’s an example of an argument where the content of the conclusion is beyond that of the premises:
All As are Bs.
All Bs are Cs.
Therefore, all Gs are Qs.
See what I mean? The premises are about As, Bs, and Cs, and the conclusion is about something else entirely different—Gs and Qs. The problem is the argument is invalid. It is certainly possible for the premises to be true and the conclusion to be false. Now, the last argument is just plain silly. But, consider another argument where the conclusion is about something that is not contained in the premises. Suppose I’m interested in finding out what percentage of NFL football fans in Columbia, Missouri, root for the Kansas City Chiefs as opposed to the St. Louis Rams (as you may know, Columbia is located equidistant from St. Louis and Kansas City). I don’t feel like calling all NFL football fans in Columbia (there are probably tens of thousands of them). Instead, I’ll make 1,000 calls. Suppose the result is that 58% of the 1,000 NFL fans I called said they root for the Rams. I think I have enough to formulate an argument with one premise and one conclusion:
1. 58% of the 1,000 NFL fans I called said they root for the Rams.
2. Therefore, 58% of the NFL fans in Columbia, Missouri, root for the Rams.
Now, let’s assign variables to the argument and put it in deductive form to see whether it is valid (which it is not).
A = 58% of the 1,000 NFL fans I called said they root for the Rams.
How should I translate the conclusion into a variable? Well, I certainly cannot assign A again because the conclusion says something other than A. The conclusion says that 58% of all (as opposed to 1,000) Columbia NFL fans root for the Rams. So, I need to assign another variable. This already gives you another sense in which the conclusions of some arguments go beyond what is contained in the premises. Here’s the argument form:
2. Therefore, B.
Clearly, this is an invalid argument.
Before I go on to ask whether the argument is any “good,” I want to make a subtle point. Students often point out that the sample size is small (1,000 out of probably tens of thousands of Columbia NFL football fans). But, I want you to see that the sample size makes no difference to the validity of the argument. Even if I call every NFL football fan in Columbia, the argument would still be invalid. Why? Because the argument form (A; Therefore, B) is invalid. It is possible to imagine an argument of the same form where the premises are true and the conclusion is false. Even in the case where I’ve called every NFL football fan, you can imagine that it is possible that someone is lying; hence, this is a case in which the premises are all true and the conclusion is false. Bottom line: no argument based on an empirical sample is ever valid (now, that’s profound!).
Here’s the next question: is the argument any good? More generally, are arguments from empirical samples
“good”? We’re going to need to be subtle about this and think this through very carefully. If we define “good arguments” in terms of a guarantee for truth, then, by definition the argument is no good. I think this is a common reason for students to reject arguments from empirical sampling (especially after they have gone through all the lessons of validity unscathed). But, suppose we expand our definition of a “good argument” to include good reasons for believing a conclusion.
Now, is the argument any good? That is, do we have good reasons for believing that 58% of Columbia NFL fans root for the Rams, given the poll I conducted? Does it make a difference to you if I significantly increase the poll size from 1,000 to 60,000 (assuming there are that many NFL fans in Columbia)? If you are a holdout and refuse to believe that the argument is good, even if I increase the poll size, then I’ll ask you to consider what would make a good reason to believe in the conclusion that 58% of the NFL fans in Columbia root for the Rams. What kind of argument would convince you of the truth of the conclusion? As I said, no argument from empirical sampling is ever valid; hence, there is no way to make a valid deductive argument to support the conclusion. The desire of a population of people is not something that you can know by definition or by logical reasoning. You have to ask people what they like and dislike. So, it seems that we can never guarantee the truth of the conclusion about people’s NFL team allegiances. All such conclusions would be supported by invalid arguments. But, still, it would be unreasonable to suggest that no poll data or sampling data could ever give
us some good reason for believing the conclusion inferred. Here’s what I’m trying to convince you about arguments from sample data: while in our argument about who roots for the Rams, the premises do not provide a guarantee that our conclusion is true, they do provide us with something weaker but still worthwhile—a good reason for thinking the conclusion is true, albeit believing the conclusion is a little risky.
The argument from sample data is called an inductive argument (http://en.wikipedia.org/wiki/Inductive_reasoning). Distilling everything I just told you about the difference between inductive and deductive arguments, the bottom line is that inductive arguments are never valid and are not intended to be valid. So, we don’t evaluate them in the same way. In particular, the arguments about NFL fans are what the authors call “statistical generalizations” (pages 219–220). The example the authors use concerns Canadian quarters, but the lesson learned is the same as I provided in the commentary. How do we evaluate inductive arguments? We say they are to some degree strong or weak. In general, there are two ways to evaluate the strength of an inductive argument.
First is sample size (pages 220–222). The greater the percentage of Columbians we poll, the more confident we are that our poll results reflect the truth about Columbian NFL allegiances. We encountered this before. Some students are not confident of the conclusion that 58% of Columbians root for the Rams because I haven’t called enough people (1,000 is too small of a percentage). Their comfort level with the conclusion is increased the greater the sample size.
Though beware: no matter how many people are sampled, there remains a possibility that further information can undercut the argument (e.g., that some of the respondents to my poll were lying). This constitutes a difference between inductive and deductive arguments. The validity of a deductive argument is never lost by adding more premises. Philosophers say that for this reason, valid arguments are indefeasible, while inductive arguments are defeasible.
A second way to evaluate the strength of an inductive argument is by determining whether the sample is biased (pages 222–224). Suppose I were to call 10,000 as opposed to 1,000 Columbians, with the result that 78% said they are Rams fans. At that point, you might say, “See, I told you—the sample size makes a big difference!” The change of percentage of Rams fans increased dramatically as I increased the sample size tenfold. But, suppose
you were to inquire a little further about the source of my sample (perhaps you are a little skeptical that the percentage would change so drastically). Suppose my answer is that I found all the Columbians who have signed up for more information from the St. Louis Rams Web site (http://www.stlouisrams.com/splash/). Aha! What does this tell you about the reliability of my sample data? Since the list comes from a Rams fan Web site, it isn’t a fair sample of all Columbians. That is, the data are skewed. So, the inference from the poll data (where 78% say they are Rams fans) to the conclusion “78% of Columbians are Rams fans” is very weak.
You should notice by now a further difference between inductive and deductive arguments. An argument is either valid or not. There is no in-between. But, the strength of an inductive argument is a matter of degree depending (in sum) on the sample size and biasness of the sample.