# How likely is the story to be true in the first place?

## The Sniff Test, Question 2

Suppose you visit your doctor and are given a routine test for a certain form of cancer. Suppose that about .5% of the population has this cancer. Your test results come back, and much to your horror, the test is positive. The cancer test has 99% sensitivity, which sounds pretty trustworthy (99% of the time the cancer is present, it will detect it). Does this mean there’s an 99% chance that you have cancer? Would you begin to worry like George Costanza that you’ll die an early death? Would you call your lawyer and have him draw up your will? (Note: I’ve changed some assumptions of the story from the original paragraph to avoid an earlier calculation error.)

This would be hasty for many reasons. One is that you might be committing a common fallacy that probability theorists call the base rate fallacy.

At this point, you know three things. First, only .5% of the population gets this form of cancer. That is the prior probability of a random person’s getting this kind of cancer. Second, you know that this one cancer test is positive. Third, you also know that even though the cancer test is 99% sensitive, it also delivers false positives. Let’s say that 99% of the time, someone who is not sick gets a true negative. But 1% of the people who aren’t sick get a false positive.

So you started off with no evidence that you had cancer, and now you have a positive result from a test that is 99% sensitive to a very rare form of cancer. This is definitely real evidence, but given the rarity of the condition in the first place, probability theorists would calculate your new odds of having cancer at still only about 33.2%, which is still low. Even if only 1% of people who aren’t sick get false positives, because the cancer is so rare, there are a lot of people without cancer who will get false positives.

(I’ll talk about the math here in the brief appendix. Note: Because I changed some assumptions in my original first paragraph, the calculation of probability in the above two has also changed, because there was an embarrassing error in my original calculation. But the correct calculation still illustrates the point.)

The base rate fallacy is essentially the fallacy of the hypochondriac. . . . [It] is also the fallacy involved in believing an improbable news story without further confirmation.

Cancer is so unlikely to begin with that much more than a single test will be needed to prove that it’s there. And of course that’s what you’re going to get! No doctor worth his or her salt is going to give you one test and begin treatment. Prepare for a battery of tests. The moral of the story is the old adage: extraordinary claims require extraordinary evidence.

The base rate fallacy is essentially the fallacy of the hypochondriac. Discovering that they have one symptom of a terrible disease, their mind races to take seriously that they have this highly unlikely disease. But as they teach first-year medical students, when you hear hoofbeats, think horses, not zebras. That fever is much more likely due to influenza than it is to fatal Rocky Mountain spotted fever.

The base rate fallacy is also the fallacy involved in believing an improbable news story without further confirmation.

In Part 1 of this series, I suggested that there are some simple questions we can ask ourselves about the stories we encounter online. So far I’ve suggested that we should first think about what we know, if anything, about the source. We should try as best as we can to accept reports from only reliable sources. Of course even reliable sources can get things wrong. (“Reliable” does not mean “infallible.”) So even when a story passes the first step of the sniff test, we still need to run other tests. The second question of the sniff test is: How likely is this story to be true in the first place? In other words, a key part of running the sniff test on a story is to consider the prior probability that a story of this type would ever occur.

According to at least one survey, one of the most shared “fake news” stories several months before the election was one with the headline, “Pope Francis Shocks World, Endorses Donald Trump for President, Releases Statement.” Nearly a million people engaged with this story (whether by sharing it, commenting on it, or reacting to it on Facebook in the months leading up to the election).

Shocking this would be, indeed. So shocking, that one should consider just how likely it is to be true.

Suppose just for the sake of argument that you heard this story from a known reliable source. Even then, the odds that Francis would do such a thing are so abysmally low in the first place that even after hearing the story your estimate of the probability that it is true should remain relatively low. Anyone who has followed the election even only passively should realize that these are people with very different personalities and philosophies. For that matter, the Catholic pontiff does not usually endorse political candidates from either party. People who follow the election more actively would remember that the two of them even got into a tiff earlier in the election cycle.

Even hearing this from a reliable source would give you no better odds than that first reliable cancer test did. So it’s adding insult to injury to consider that people believed this story when it appeared originally in something called WTOE 5 NEWS: YOUR LOCAL NEWS NOW.

This step of the sniff test was especially useful this election season in avoiding the temptation to believe allegations about the nasty words and deeds of the candidates. I am no fan of Donald Trump, but I knew that most of the fake tweets attributed to him would turn out to be fake. Even if Trump is completely evil, he isn’t completely stupid, and he knows that he can’t say anything and get away with it. The same goes for many of the nefarious statements that have been attributed to Hillary Clinton and Obama over the years.

The foolishness of believing improbable stories without further confirmation is at least somewhat excusable when someone is young and doesn’t know much about how the world works. It is completely inexcusable when someone has been around for many decades and should be able to remember the last election or three.

Of course improbable things sometimes happen — like the election of Trump. And that’s why I’m only describing a sniff test, not a comprehensive diagnostic procedure. If a story is improbable to begin with, early indication that it is true should at most prompt you to search for confirming evidence — just like that initial cancer test. And that’s when the initial indication is reliable. Not even the mainstream media is 99% sensitive to the truth. I will leave it as an exercise for readers to consider what to do when the story comes from an “alternative” media source.

Given the political incentive and the technological ease of spreading made-up stuff, it would be a miracle if it weren’t spreading like wildfire in an election season.

The 18th century Scottish philosopher David Hume used similar probabilistic reasoning to disqualify the reliability of testimony about supernatural miracles. He thought that because miracles by definition involved the violation of a law of nature, believing reports of miracles would contravene all of our past experience. But, as I mentioned in my earlier post, he thought our primary reason for trusting the words of others is our prior experience of the accuracy and veracity of testimony. To the extent that we must discard our past experience to believe reports of miracles, we undermine the whole basis for relying on testimony in the first place — including testimony of the miraculous. Hume’s verdict was telling:

When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle. If the falsehood of his testimony would be more miraculous, than the event which he relates; then, and not till then, can he pretend to command my belief or opinion. (An Enquiry concerning Human Understanding, 1748)

By analogy, whenever anyone tells me that the Pope endorsed Donald Trump, I consider whether it is more probable that this person should either deceive or be deceived, or that the Pope really did endorse Trump. A papal presidential endorsement of a populist Presbyterian reality TV show host would seem miraculous enough. But it would be no miracle for people to be making this stuff up. For both sides of the political spectrum, and for most of American history, there has always been an incentive to spread lies and exaggerations to make one’s own candidate look good. The only thing new about “fake news” on the Internet is how easy it is to produce and spread. Given the political incentive and the technological ease of spreading made-up stuff, it would be a miracle if it weren’t spreading like wildfire in an election season.

#### Appendix (updated to correct for a miscalculation)

I’d like to describe briefly how that 33.2% chance of having cancer was calculated earlier in the post. Readers whose eyes glaze over with math (like mine do) can stop now. But push on a little further if you want an extra tip for running the sniff test. To get the 33.2% number, I was using Bayes’ theorem, which derived from the work of the 18th century mathematician Thomas Bayes:

You can see an example of the how the 33.2% number is calculated on the Wikipedia entry on Bayes’ theorem, where instead the numbers apply to a hypothetical example about drug testing. I wanted to talk about cancer testing instead because of the connection to the hypochondriac.

The equation might look a little daunting, but it’s actually a pretty simple idea. p(H|E) is the probability of the hypothesis H you’re trying to determine, based on new evidence E (e.g., the chance that you have cancer, given that you’ve had a positive cancer test). p(H) is the prior probability of the hypothesis (e.g., the .5% base rate of cancer), p(E) is the probability that the evidence would appear (whether one has cancer or not), and p(E|H) is the probability that this evidence would appear supposing the hypothesis were true (for instance, 99% of the time there is cancer, the cancer test shows it, even though it also shows false positives when there is no cancer).

I don’t know that Bayes’ theorem captures the idea of probability conditional on new evidence perfectly, but it certainly captures some important ideas about probability. It captures the idea that the probability of the hypothesis given newly discovered evidence increases with increasing p(E|H) and increasing p(H), but decreases with increasing p(E). So if p(H) is really low (like .5%), then even if p(E|H) is really high (like 99%), the overall probability function could still be low.

But of special note is the inverse proportionality of p(E). Think about this with reference to the fake news phenomenon. If the Pope really did endorse Trump, there’s about a really good chance that a whole variety of conservative news sites would report this, making P(H|E) very high. At the same time, p(E) is also relatively high, which counteracts that high reliability. That’s because you know there is fake news and many people who will be making stuff up because it would be pleasantly shocking if true to many people. There’s probably no good way to calculate the odds that someone would make up this particular story in advance, but the point is that the more likely that someone would make up some evidence, the less likely it is that observing it will make your hypothesis true.

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