Skeptics, Logic and Probability

Human beings, who are almost unique in having the ability to learn from the experience of others, are also remarkable for their apparent disinclination to do so.
— Douglas Adams, from Last Chance To See

Sceptics are those people who, like me, do not accept extraordinary claims without the extraordinary evidence such claims require to make them believable.

Sceptics are also open-minded, despite the wailing of self-professed experts and other believers in all things paranormal. Those “experts” in woo think that open-mindedness means having a willingness to believe in the possibility that their claims are true.

But there is another definition of open-mindedness. Open-mindedness  means simply the willingness to change one’s mind in the light of new evidence. Believers in the paranormal almost always  fail that test, because no matter how often they fail to prove their claims, and no matter what evidence they come across that contradicts their beliefs, they cling tenaciously to claims they cannot prove, and which are often disproven. The closed-minded people are the ones who cannot or will not alter their views when new evidence comes along.

Scepticism itself is not a profession as such, but anyone can doubt claims that are made without evidence – and that’s the way it should be. Anyone who believes in ghosts or psychics or any other unproven claim on weak to non-existent evidence is not being open-minded, they are being gullible. When someone makes assertions about the reality of the paranormal, it is only right that others should ask, “What’s your evidence?” Or just the outright challenge: “Go on, then – prove it.”

Unfortunately, very few of the believers in – and promoters of – the paranormal have any understanding of logic; they assume that because their own reasoning makes sense to them, then they know something about logic. But logic is not a matter of common sense. Like much of science, there are many aspects of logic that are counter-intuitive. It is a subject that has to be studied – preferably under the guidance of someone who is qualified to teach it, or at least has passed the relevant accredited examinations. Not many people do study logic in a formal way, and that probably explains why so many believers commit so many fallacies when they are trying to support their beliefs with what they think is a logical argument, but what is, in fact, wishful thinking, rhetoric and sometimes pure sophistry. They might be genuine in their beliefs, but their arguments are wrong and they usually don’t know it, and they certainly don’t know why their arguments don’t hold up.

One of the most naive arguments put forward by the “experts” is often along the lines of: “An alleged paranormal event witnessed by numerous people should be given more weight than a similar alleged event witnessed by just one person.” No, it depends on whether the alleged event has any “prior plausibility.” It also depends on whether the alleged event has any independent confirmation.

Logic, however, is a theme that underlies this blog anyway, so I’m not going to make this post just about logic; I want to bring in a related theme – probability.

I thought about it recently when I read this book review by Harriet Hall. The book is: Dicing With Death: Chance, Risk and Health by  Stephen Senn. I am certainly going to get a copy in the near future. Essentially, it is a book about statistics, and how probability relates to so many things in the everyday world. Woomeisters often throw out spurious statistics to try to support the nonsense they spout,  so it’s a good idea to be aware of what can and cannot be justified with numbers.

Harriet Hall gives an example of probability from the book, and although she gives the answer to the problem, she does not explain why it is the answer. I assume that was a teaser to encourage people to buy the book, but I thought I would give the explanation here anyway. What interested me most was the fact that it provoked some discussion in the comments.

Here’s the problem:

If a man has 2 children and at least one of them is a boy, how likely is it that the other is a girl? Most people reason that there are only 2 possibilities, boy or girl, both equally likely, so there is a probability of 1 in 2, or 50%, that the other child is a girl. That’s wrong. In fact, there is a probability of 2 in 3: the other child is twice as likely to be a girl as a boy. The 50% answer is only true if you change the question slightly from “one of them is a boy” to “the firstborn is a boy.” If this doesn’t make sense to you, you really need to read the book

In logic (and most of science), common sense is not a good guide to what you might think is going on, and that applies also in statistical analysis. And it’s also the case that the set of numbers arrived at when an analysis is done might well need some degree of interpretation. If a psychic, for example, scores higher than chance expectation in a series of tests for psi, does that mean that psi has been proven? So far, no. But interpretation of results doesn’t mean forcing them into any preconceived belief you might already have. Some logical analysis also has to be applied in order to justify one’s final conclusion. (If one third of road accidents involve drink-drivers, does that mean that sober drivers, who are in twice as many road accidents, are the ones who should be banned from driving? If not, why not? Discuss.)

Statistics, as a subject in its own right, can become very complicated, depending on how far you might want to get into it. But for the purposes of this post, it’s enough to just deal with some of the basics – in essence, statistics is about probability.

But back to the puzzle. You can take it that boys and girls are born with the same 50/50 probability. There’s no need to worry about other factors like the rare occurrence of hermaphrodites or children who later become transgender, unless the problem specifically includes that information. In these kind of puzzles the information necessary to solve them is provided without any other assumptions having to be made.

But why, out of two children in this particular  family, is there a probability of 2/3 of the other child being a girl?

Consider it this way: in a two child family, there are four possible combinations of births: boy/boy; boy/girl; girl/boy; girl/girl. You already know that at least one of the children is a boy, which rules out the girl/girl combination. Out of the three remaining possibilities, one possibility is that there are two boys, but there are two possibilities that include a girl. So the chances of the other child being a girl is 2/3. Counterintuitive, but true.

If the problem had stated specifically that, say, the first born had been a boy, then the probability that the other child was a girl would, indeed, be 50/50.

But there are some probability problems that are so counter-intuitive, that even mathematicians have been in vehement disagreement with each other. One of the most famous of these problems has become a classic of its kind: the so-called Monty Hall Problem.

Monty Hall hosted a US TV game show called Let’s Make A Deal. The highlight of the show came when the leading contestant had the chance to win a big prize – maybe a car – or a pretty worthless booby prize. It worked like this:

Monty Hall presented the contestant with three doors, one of which had the star prize behind it, but the other two had booby prizes behind them. The contestant was then invited to choose the door he guessed might hide his new car. At this stage, only Monty Hall knew which door hid the prize; it could be the contestant’s original choice, or it could be one of the other two doors. In any case, Monty Hall would then open one of the other doors, showing that it was not the star prize. The contestant was then offered the option of changing his original choice, and select the other closed door instead.

Here’s the problem: should the contestant stick with his first choice, or does he have a better chance of winning if he switches? Or doesn’t it make any difference? There are only two choices now so is it just a 50/50 chance of winning, or does he increase his chances of winning if he switches?

Strangely enough, the contestant will double the probability of winning the star prize if he switches. If he does so, his probability of winning goes from 1/3 to 2/3.

I’ve had some interesting conversations with people who will not accept that answer, because they see the problem as changing a 1/3 probability into a 50/50 probability and as far as they see it, 2/3 just doesn’t ring true when there are now just two choices.

Here’s why the contestant will double his chance of winning if he switches:

Suppose he chooses door A. If that was his only choice, then his chances of being right are 1/3.  Each of the other doors, B and C, also have a probability of 1/3, so together, the probability that the prize is behind one of the other two doors is 2/3. If there was no option to switch, then the contestant’s probability of winning is just 1/3. But by sticking with his original choice, it is still 1/3.

Given the fact that there is a 2/3 probability that the prize is behind one of the other two doors, it makes sense to switch. If that doesn’t seem obvious, think of it another way: imagine you had the choice of picking the prize from a million doors. You therefore have a one in a million chance of picking the prize, and 999,999 chances of being wrong. So Monty says, “OK, I’ll open 999,998 doors that do not have the star prize behind them.” What would you do then? Your original choice is still one in a million, but the probability of the prize being behind any of the other doors, including the one still unopened door is nearly (but not quite) certain. There are no guarantees, of course, but in that scenario, it would be foolish to stick with your original choice.

One other way of looking at it is if Monty Hall didn’t open any of the other doors but simply said that you can have whatever is behind all of the other doors if you give up your first choice. In the game show you get two other doors, or in the hypothetical million door choice you get 999,999.

These kind of puzzles are fun to do, but they are the basis of statistical analysis, which is so important in so many aspects of our daily lives. Medical research in particular depends on statistical analysis to work out whether new drugs are not just effective, but also safe. And those new drugs need to have a very high probability of working as expected, but also a very low probability of causing any harm because of any side effects they might have. Statistical analysis of properly controlled tests is one of the things that leads to treatments that give us longer and healthier lives than our ancestors could ever have dreamed of.

Most people, of course, are not statisticians, and we have to rely on the professionals to work out the fine details of things – like drugs – with regard to whether they work and are safe. In the everyday world, however, I think it is worthwhile for people to get to grips with some basic probability theory. It’s the misunderstanding of how likely something is that leads to the belief in many aspects of the paranormal.

A couple of years ago, for example, I followed a thread on a pro paranormal blog in which it was stated that a very small number of people had won two or more lottery jackpots. The basic idea was that because such a scenario was so unlikely, then there must be some underlying psychic activity going on that caused those winners to attract the wins they achieved.

The reasoning went like this: the probability of winning the jackpot in a 6 out of 49 draw is approximately one in 14 million. To win two such draws – even three or even four jackpots brings the odds against to astronomical levels. The odds are so unlikely, that there must be something else (psychic powers) at work. (Try working out on your pocket calculator 14 million x 14 million x 14 million, etc.)

The thing is, though, if someone is lucky enough to win multiple jackpots, probability predicts that such a scenario is inevitable. What probability theory does not predict is who the winner will be. If a lottery draw is truly random, then you cannot expect numbers to be drawn with any kind of predictable outcome, which is why there is no such thing as a “winning system” that anyone could devise.

The bad thinking going on by paranormal proponents is to focus on an unusual occurrence after the event. Basically it is a version of the Texas Sharpshooter Fallacy – taking a few pot shots at the side of a barn and then drawing bullseyes around the hits. Psychics do not routinely win the lottery, after all, and no one has ever taken a Nostradamus verse and predicted an important event before it happens.

If someone does have two or three lottery wins, so what? Unless it is predicted before it happens, there is no need to assume anything paranormal is going on. (If it did happen, though, I think the first thing the lottery people would do would be to check their system for malfunction or tampering.)

Not everything can be reduced to numbers, though. Inductive logic is also about probability, but quantifying the probability of some things is not straightforward. Are UFOs, i.e., alien space ships, real? Without knowing all the parameters, a numerical figure can’t be worked out, but the probability of alien visitation to this planet is almost zero. But am I justified in making that assertion – or are the UFO “experts” right with their claims that UFOs are here and are abducting humans on a daily basis?

Given the fact that the laws of physics apply all over the universe, I think it likely that life exists elsewhere. It’s not certain, just likely. Perhaps there are planets that even have intelligent life with science and  technology. It’s happened here, so why not elsewhere? I don’t argue that life doesn’t exist somewhere else, just that they are almost certainly not here. The laws of physics give good reasons to believe that alien visitation is unlikely in the extreme. But the UFO buffs don’t do their case any good by speculating about wormholes, other dimensions or anything else they can come up with. Physicists themselves speculate about such possibilities, but they also offer good reasons why such speculative ideas are likely to remain nothing more than that – speculation.

Given the choice between unsupported assertions by UFO believers and what science has to say, I think the best bet is to go with the science. The burden of proof is still on the people who make the claims about extraterrestrial visitation. Believers see UFOs; astronomers see meteors, space debris and other explainable phenomena. The sceptics among us just see things we can’t explain, without feeling the need to make up something to fill in the gaps. If you don’t know what it is you are seeing, there’s no shame in just saying, “I don’t know.”

When the believers make extraordinary claims without tangible evidence to support those claims, you are faced with the possibility that the claims are true, or that they are false. If the evidence isn’t there, then the claims are probably false. The possibility of aliens being here is not that they are, or they are not, and therefore it’s 50/50; the probability of aliens breaching the laws of physics is extremely unlikely, so the probability of it being true is so remote you can forget it. It’s still up to the UFO (or any other) believers to prove their case.

Most of us don’t have to do mathematical calculations in our everyday lives, but there are many instances where we do make “intuitive” calculations about what is going on. And many people go terribly wrong when they do so. Do psychics, dowsers, faith healers and all the rest of them really do the things they claim to do? Not under controlled conditions, they don’t. So they probably aren’t real.

I chuckle inwardly when some self-professed expert in the paranormal/supernatural/UFOs and all the rest of it claims that aliens, for example, are here because people say they have seen them or their space ships. It gets no better when certain astronauts claim that they have perhaps seen aliens and their ships in some secret hangar somewhere. And it becomes ludicrous when it is realised that not just a handful, but millions of Americans claim to have been abducted by the “Greys” and undergone experiments aboard strange craft, and even been subjected to sexual intrusion with the purpose of producing alien/human hybrids.

What’s the probability of that? Given the complete absence of any confirming evidence, the probability is (approximately) zero. Humans can’t successfully mate even with other animal species that evolved on this planet, so why would it be possible to mate with an alien species that would probably not have anything similar to the DNA that does at least link all life on Earth?

One thing the woomeisters have going for them is that they can spout any drivel they can think up, safe in the knowledge that nothing they invent can be disproved. I can tell you that I have fairies at the bottom of my garden and you can’t prove me wrong. I can think up an excuse to counter any objection you can think of. Want to see them? No, they’re invisible. Want to set some kind of trap to catch them? No, they’re immaterial. And so it goes on. If I make such a claim, then do  all my excuses for not providing evidence to support that claim make it any more likely that the claim is true? Only the most gullible would go along with it.

But in the face of a claim that has nothing to back it up, one can work out a rough probability calculation, even if it can’t be quantified numerically: “You make a claim with nothing to verify it? OK, I will accept its validity in proportion to the amount of testable evidence you supply. No testable evidence, no belief from me.”

That’s the problem, of course. A lot of people make money by writing nonsense they might even believe themselves and claiming that there is no onus on them to prove the claims they make; rather, that their critics have the responsibility to prove them wrong. Which is wrong.

There are other problems, of course: if something allegedly paranormal is presented on TV as a “documentary,” does that make it more likely to be true? For some people it does; a lady I know is convinced that psychics solve crimes. After all, there are TV documentaries about crime-busting psychics and they “couldn’t put it on TV if it wasn’t true,” could they? For this lady, the probability that psychics solve crimes becomes a certainty in her own mind just because it is on TV.

These “documentaries,” of course, are nothing more than dramatised re-enactments of claims made without any proof. It’s sad that people put so much faith in what they see on TV without stopping to think about it. There’s a huge difference between a real documentary (anything by David Attenborough, for instance) dealing with matters of fact, and the mindless drivel churned out for the credulous who will not question what they see.

It comes down to this: if a claim of the paranormal is presented without credible evidence, it might be true, but it probably isn’t. When a TV station presents paranormal programming, it is probably chasing ratings, and advertisers are probably pleased with the results – at least until the ratings start to fall.

If it were true that TV companies are only allowed to produce documentaries that they can prove are true, then there would probably be no more paranormal programmes to watch.

Few things can be claimed with certainty, but for everything else we have to work out some kind of probability rating. Whether it’s the lottery or any paranormal claim, many people would benefit from spending time learning some probability theory, or at least finding out what is plausible or not when it comes to deciding whether there is anything in the (actually) implausible claims made by paranormalists. Unfortunately, that isn’t going to happen any time soon.

(Additional note for those who want to know how the lottery odds are worked out – the probability of winning the jackp0t in a six out of forty nine number draw is:

1/(49 x 48 x 47 x 46 x 45 x 44 / 6 x 5 x 4 x 3 x 2 x 1)

=1/13,983,816.)

“It could be you!”

But it probably won’t be.