# Category Archives: Probability

## Statistically Impossible? Probably Not

Paranormal proponents seem to have a thing about statistics. It’s a bit of a conundrum, actually, because they love to quote some research by some big name in parapsychology who has said that the chances of a particular psychic gaining a particular score in some test or other is in the billions to one against pure chance, and therefore telepathy or whatever must be real. At the same time, they obviously haven’t a clue what they are talking about – and just don’t understand the concepts they are trying to use to justify their belief in the paranormal.

Here’s an example. A couple of years ago I joined  a discussion on a pro paranormal blog where the author pointed to a small number of people in America who had won several lottery jackpots. The point he was trying to make was that because winning the lottery is such a long shot (one chance in almost 14 million in a six from forty nine draw) that the probability of winning two, three or even more jackpots is so unlikely that there must be something paranormal going on. Pure chance, for him, just doesn’t come in to it. His belief was that those winners must be attracting those wins in some paranormal way, even if they weren’t consciously aware of it.

Sometimes, the paranormal people will make a statement along the lines of, “This psychic scored a result in a scientific test that was so improbable that it is statistically impossible for it to be just a chance result.”

To which I can only reply, “No. You don’t know what you are talking about.”

It comes down to a simple principle: statistics deals with probability theory. The probability of something happening due to chance is measured as a ratio between zero and one. A probability of one = certainty. A probability of zero = impossible.

Statistics is used to analyse the probability of a particular event happening. If you buy a lottery ticket, then your probability of hitting the jackpot is one in (approximately) 14 million (in a six out of forty nine draw). The next week if you buy a ticket, the probability of you winning is also one in 14 million. Whether you win or lose the first draw, it has no effect whatsoever on whether you win or lose the second draw; they are independent events. The probability of you winning both draws can be worked out by the multiplication rule, so 14 million x 14 million, represents one chance in 196,000,000,000,000. How do you like those odds?

If you want to know the probability of winning the third jackpot, then multiply that number by another 14 million.  And so on. It doesn’t take long for those numbers to become mind-bogglingly big.

But here’s the big question: at what point does a particular posited event become “statistically impossible?” When can you declare that something is impossible – or that there must be something psychic going on as the only possible explanation for an unlikely event?

Ask yourself this: why should it be the case that a small number of people (who do not regard themselves as psychics) win several jackpots, but no self-professed psychic can predict the next lottery draw? It’s rather cliched, but ask a “psychic” what the next lottery numbers are going to be and he or she will squirm out of it, almost always starting with the words, “It doesn’t work like that.” How very true, but they won’t tell you how it does “work,” either.

A long time ago – more than twenty years ago, I’m sure – I was walking along the local central shopping centre and noticed that there was a stall set up by a charity. There was a new car parked there, and on the stall there was an offer: “Win this car. Throw six sixes with these dice and you win it. £1.00 per entry.”

Would you take a chance on it? OK, this was a long time ago. The car was small and possibly worth (maybe) about a couple of thousand pounds at the time. On the other hand, a pound was worth more than it is now. But even if my guess and memory are correct, there is still a huge difference between a pound and several thousand pounds at stake at the time. Would you take the risk? All you have to do is throw six dice and get six sixes. Easy or what?

But what if you represent the charity that is trying to raise money for an undoubtedly good cause by taking the risk of giving away an expensive prize to someone who might win it with the very first throw of the dice? What are the actual risks here?

The reality is this: when a charity offers a huge prize (that they can’t really afford to lose), they employ an insurance company to take the risk on their behalf. On the other hand, if it pays off, the charity stands to make a lot of money to help them pursue their charitable goals. And that’s a good thing.

In fact, they won’t lose. Even if someone throws the six sixes on the first throw, the risk will be taken by the insurance company — in exchange for a premium, of course, which will be paid for from the draw takings – and the charity takes the rest. The insurance company calculates the odds, and charges the premium the charity pays accordingly. Depending on how the actuary of the insurance company works it out, there might, for instance be a limit imposed on how many dice throws are allowed during the charitable event. Or the premium to be paid might be adjusted for the number of throws over and above the basic calculation.

I’ll just add a note here: the charity will have to pay the premium, even if someone does actually win on the first (or a very early) throw. There’s still a risk, even for the charity, but the risk is small and in any case, it is an “allowable expense,” as it were.

I’ll add another note just for interest: a charity will also have a good relationship with the press and the event will be well publicised – which means that the additional publicity will also attract donations from people who support that charity’s aims. Those donors won’t have the opportunity to try to win the car, but will donate just because they are nice people who are willing to be altruistic. Good for them.

But look at it this way: if the car in the event I mentioned was valued at, say, £2,000 at the time, and participants were paying £1.00 per throw, there are 46,656 (6x6x6x6x6x6) possible combinations that the six dice could fall on. Although the first participant could throw the winning combination, he probably won’t. It’s also possible that even if exactly 46,656 people took part, there would still be no winner.

Double that number to 93,312 participants, and there is still no guarantee that there would be a winner. Or perhaps the winning throw would happen on the very last throw. When it comes to pure chance, the outcome cannot be predicted. An insurance company might take the risk, but overall the odds are on their side. Occasionally, of course, an insurance company has to pay out, but how often do you hear of an insurance company going out of business because they got their sums wrong? And keep in mind the fact that some high risks are farmed out, as it were, to other insurance companies to share the risk. Insurance is, in fact, a highly profitable business. It’s for the same reason that you never see a down at heel bookmaker – they do the same thing with big risks.

To be realistic, a charity event like the one I’ve already mentioned being played over the course of one day in a busy shopping centre might attract a few hundred people (rather than tens of thousands) to part with their money. In that case, it’s unlikely that anyone will win the big prize, so the punters have a cheap flutter; the insurance company collects its premium; the charity adds a few hundred pounds to its funds and everyone is happy.

Would you say, though, that odds of one in 46,656 are impossible? No, of course not. Obviously some people win the lottery jackpot and the odds are even longer. Neither are “statistically impossible.” The same applies to multiple lottery wins. The point is this: if the probability of a particular event is greater than zero, then that event is certainly possible, however unlikely it seems to be. There is no such thing as a “statistical impossibility,” but some things are “statistically unlikely.” If I pay a pound to try to throw six sixes in that charity competition, I probably won’t win. And that’s the same reason I don’t play the lottery, with its even longer odds. (I do, however, donate to charities, so I’m not being a skinflint for not entering the prize competition.)

Unfortunately, probability is something that most people just do not have an intuitive grasp of. One person I met some time ago claimed that he played the lottery because he reckoned he had a fifty-fifty chance of winning; his reasoning was that you can only win or lose, ergo a probability of 0.5. He is, in fact, a gift to the gambling industry but I don’t think he will ever realise it.

So what does it mean when the pro paranormalists claim that some psychic scored a result that was “statistically impossible” in a test for telepathy, or whatever? Actually, such a claim means nothing at all. If the probability (greater than zero) of an event can be calculated, then it is possible. Only a calculated probability of zero is impossible. Think of it this way – what is the probability of throwing six sixes with only five dice? Zero, of course. But what is the probability of someone flipping a coin to land heads up with a double-headed coin? The probability is one (it’s certain).

So what is it that the pro paranormalists are getting at when they say so confidently that some psychics really have paranormal powers “because their score in a test was statistically impossible?” One thing is sure: no one in the paranormal field ever gives a figure – a probability level – that they can prove mathematically that marks the dividing line between possible and impossible. If a probability of one in a million is proposed, can that be said to be the dividing line? Maybe not; after all, some people win the lottery against much bigger odds. And some people have won several lotteries, remember.

Try this idea: suppose a psychic were to claim that if he were dealt all the cards in a well shuffled (randomised) deck, he would, using his powers, “attract” those cards in a specific order – Ace to king of Diamonds, then Hearts, then Clubs followed by Spades.Would you think he had the powers he claimed to have? (I might, but only under certain conditions that I will come to later.)

If you were dealt all the cards in a deck but you got a seemingly random hodge-podge of cards, would you think that you had attracted them through some kind of latent psychic powers you were not aware you had? I’m relating here to the hypothesis of the above blogger who assumed that because a very small number of people had won more than one lottery jackpot that they must have psychic powers as the explanation for their good fortune. Could the millions of people who lose out regularly be attracting bad luck through equally effective, but “negative,” psychic powers of their own?

Look at the card-dealing scenario again. Here’s something that might surprise you: the probability of being dealt any particular sequence of cards from a full deck  is one in nearly ten to the power of 68. In other words, there is a mind boggling number of possible combinations of cards that you could be dealt. If your hand is fair (the deck is not rigged in any way, and the deal is truly random) then any possible combination of cards could be dealt – including our psychic’s four straights. In a random draw, that combination is no more or less likely than any other combination.

Here’s what makes the difference: the blogger I referred to made his decision about the probability of a small number of people winning several jackpots after they had already won. That’s a fallacy called The Texas Sharpshooter Fallacy (drawing the bulls-eyes around the bullet holes after the event. I’ll do a post about it later.)

The same applies to our putative psychic who tells you before his cards are dealt that he will attract the consecutively numbered cards of each suit. But before the cards were dealt, he could have claimed to attract any particular sequence of cards out of 10^68 possible combinations, not just a particularly memorable sequence. To put it bluntly, no combination of cards is any more or less likely than any other – including Ace to King, or any other combination you can think of. When you are dealt such a  hand, the odds of getting those particular cards is 1 in nearly 10^68 – a one followed by sixty eight zeros.

Even with such a huge number of  possible combinations, however, you must get one of them. The odds against you getting those particular cards are incredible.

What makes the difference is whether or not a psychic predicts what will happen before it happens, or whether an unlikely event is claimed to be paranormal after it happens. The same applies to the so-called prophesies of Nostradamus – every claim made that he predicted this or that event is made after it happened. No one has ever successfully  identified an event supposedly predicted by him before it happened.

So would I really believe that someone is psychic if he or she predicted an unlikely outcome before it happened? It depends on several things:

First, there would have to be strict controls in place. If a psychic intends to predict a sequence of playing cards, then he does not get to be the dealer. Nor does he provide his own cards.

Second, the cards would be a brand new, sealed pack, which would then be randomised by an independent person (using a shuffling device of some kind that does not rely on his own randomisation process) who does not take any further part in the experiment.

Third, the cards would be dealt by some mechanical means so that no further human contact is involved.

Fourth, there are other controls I can think of, all of which would be designed to ensure that there is no possibility of either fraud or trickery.

Fifth, I would have to keep in mind that no experiment is perfect; there might be something I have missed. The best an experiment can do is reveal that its result is probably not due to chance alone, not that a result is “impossible” by chance alone.

Sixth, to the believers in psi who would say that such a setup “is not conducive to the production of psychic effects,” I would say: “Learn some science and stop whining.”

If our psychic successfully beats incredible odds, can we now declare him to have psychic powers? Unfortunately not. In science a single experiment is not definitive. Even at those huge odds, a fluke can still happen, which is why the experiment would have to be repeated many times. It would also have to be confirmed by independent researchers; in other words it would have to be replicated by others who are not involved with the original research.

Parapsychologists always seem to fall down at that hurdle. Whatever the alleged psychic phenomenon is, their experimental results only seem to happen for them, in their own laboratories. No one is suggesting that any of them are cheating in any way, but other researchers often find flaws in their methodology, and even if their experiments seem OK and are faithfully copied, the same results just fail to materialise.

There is also another problem with researching paranormal claims. The fact that an experimental result shows that a claimed paranormal event is probably not due to chance does not imply that the event is actually paranormal. Unfortunately, paranormal research has been plagued with fraud – so much so that scepticism is justified; some would say it is a requirement. Even some of the best researchers in the past (and some in the present) have been hoodwinked by various charlatans – and some have been charlatans themselves. But even without any kind of cheating, there might always just be something that the experimenter has overlooked. No experiment can ever be claimed to be perfect.

I would add this: if a professed psychic were to beat huge odds against, shall we say, information transfer by pure chance – at whatever probability level is set, I would accept that that information transfer is probably not due to pure dumb luck. I would think that the information transfer has happened, but there are many ways that that could happen without the need for a paranormal explanation, or an accusation of fraud.

Given the fact that even skilled paranormal researchers have failed to prove the existence of the paranormal, what credence should be given to other claims – like the one above about a small number of individuals winning several lottery jackpots? The simple fact is this: coincidences happen. Far from it being “statistically impossible” for someone to win multiple lottery jackpots, it is, in fact, a statistical certainty that some people will do just that. The big problem lies in predicting a coincidence before it happens.

The important thing to keep in mind about all this is that if a probability can be calculated, then that is the probability of a given event happening just by pure chance – however long the odds. A long shot that works out does not imply that something paranormal is going on. And when a statistically unlikely event is claimed to be paranormal only after it happens, forget it.

Ask a psychic to predict the next lottery draw. He or she will give reasons why they can’t or won’t do it. Don’t be surprised.

Then read some report of someone whose claim to have predicted something unlikely or unexpected surfaces only after the event, and see the paranormal folk go wild about it. Also don’t be surprised.

That’s why James Randi’s million dollars, for example, is safe. Psychics claim they can beat the odds, but if they could, the Million Dollar Challenge would have been won a long time ago.

Excuse me while I chuckle inwardly. Last time I checked, the MDC was a two part test. The probability of passing the first part by pure chance is one in one thousand. Get through that and you get to the second and final part – also a one in one thousand chance. The probability of getting through both tests by pure chance is (by the multiplication rule I mentioned above) one in one million.

Ordinary (non-famous) psychics apply for Randi’s prize and fail; they don’t have much to lose, but they do have a one in a million chance of hitting the big time. It might just work out for one of them one day.

On the other hand, the famous psychics who are raking in the cash already, have everything to lose – fail a big and famous test like Randi’s Challenge and they will be finished.

Psychics make claims about their alleged powers which, if true, would beat astronomical odds on a regular basis. I would start to believe in them if they would actually do what they claim. Wouldn’t it be nice if the evening news could have a guest psychic appear on screen half an hour before the live lottery draw to announce, “And tonight’s lottery numbers will be… “?

Or more importantly, news announcements about next week’s earthquake – and the fact that evacuation of the inhabitants is already underway in good time. Or perhaps after the earthquake those people who refused to leave in time were being found alive in the rubble of various buildings by remote viewers rather than rescue workers who have to use thermal cameras, microphones, sniffer dogs and so on.

Will that ever happen in the future? The statistical odds are against it. But maybe a psychic can tell us when it will happen?

I’m not going to bet money on that, either, but I will give the last word to Albert Einstein:

“God does not play at dice with the Universe.”

[Note: Einstein was referring to the new science of quantum physics; he wasn’t declaring a belief in any gods, it was a metaphor to illustrate that he couldn’t come to terms with the fact that subatomic events can happen randomly, without a preceding cause. His science dealt with the very big (the universe), whereas quantum physics deals with the very small (subatomic sizes). He never did manage to achieve his ambition to create a “theory of everything.”]

Claiming a coincidence to be paranormal is just bad thinking.

## Alien Invasion Might Be More Horrific Than You Thought

I found an amusing takedown of another bit of credulous UFO apologetics. Yeah, UFOs must be real because you can’t prove they’re not. Sceptics are just awful.

I’m old enough to vaguely remember the beginning of mankind’s exploration of space, starting with Sputnik, and I’ve grown up with the fantastic advances in science and technology that we have all seen and gained the benefit of. In fact, for me, this is the most exciting time to be alive, and I hope that, before I snuff it, life elsewhere in the universe will be confirmed.

Right now, NASA is sending probes out into space to try to detect life; in particular, there are machines on Mars designed specifically to find out if that planet could have supported life in the past, or even if there might be the remnants of actual life there now. And the SETI Institute is constantly scanning the skies in the hope that we might detect signals from other civilisations.

All of this is being done very publicly, and there are some excellent documentaries being broadcast that deal with the latest ideas in science that discuss the likelihood of alien civilisations being out there, and the possible ways that other life might have evolved. It’s not settled yet, of course, but the laws of physics apply all over the universe: it’s likely that life is abundant. If life could arise on Earth, there is no reason to assume that the same couldn’t happen elsewhere.

(By the way – I mentioned “excellent documentaries”: I mean things like Wonders Of The Universe, not unmitigated cobblers like Ancient Aliens or UFO Files. )

“”Ufology” – a pseudoscience if ever there was one – is like any other aspect of paranormal investigation, and totally unlike any real science. The existence of the claimed phenomenon is inferred from the fact that the paranormal investigators just can’t think of (or accept) an ordinary explanation. A light in the sky? No idea what it is, therefore it’s an alien spaceship from another galaxy (ASFAG, as I call it – not UFO, which by definition is something that has not been identified).

I’m not claiming it’s impossible for aliens to be here, just that it’s highly unlikely. If it’s true, then we need confirmable evidence, not the say-so of a few cranks who write uninformed magazine or newspaper columns, have books to sell and/or a profitable career to pursue on the UFO lecture circuit.

Despite the various science fiction scenarios we are all familiar with –- flesh-eating aliens, creatures that want Earth’s natural resources, the assimilation of this planet into some galactic empire, and so on, there could be something worse in store for us if and when these aliens do arrive. What if (shudder) these “people” have got… RELIGION?

Just think of it for a moment. Instead of spaceships with alien scientists and anthropologists on board, there might be exotic craft heading our way full of Missionaries! (Aaaargh!)

It might seem incredible to think of an advanced civilisation worshiping any gods. After all, the one thing that stands in the way of scientific progress here is religion. Some of the more liberal religious people accept much of their religion as being largely allegorical and metaphorical, without taking away their belief in a creator, which allows science to progress without too much interference (some scientists are religious, but at least they do tend to be deists rather than theists). Fundamentalists, however, will not accept science if it contradicts their religious beliefs. In fact, there are many fundie organisations that say categorically that anything that contradicts their dogma (especially science, which happens to be testable with no faith required) must be rejected. Creationists are trying to get their beliefs taught in science classes as though there is anything remotely scientific about any of it. If they get their way, then real science will be destroyed, along with any hope we might have of ever reaching the stars ourselves.

Given the history of religious strife on Earth – an enterprise that has turned countries into graveyards, the possibility of aliens being evangelicals is a thought that is quite horrifying.

Then again, aliens, if they arrive here, will (I hope) have probably ditched religion eons ago, but they might just clear off as soon as the fundies start knocking on their spaceship doors to “spread the good word” – or worse – “destroy the infidels.” The aliens might conclude that their search for intelligent life has drawn a blank in this backwater of the Milky Way.

No one knows for sure what’s out there; we can only say that alien life probably does exist, but is probably not here.

For an illustration of the probability of human/alien interaction, this is a good analysis (with thanks to xkcd):

I’m going to be optimistic on this one. If aliens still have religion, then they will very likely have done the same as the fundies on this planet have historically done, i.e., tortured and killed their best thinkers. Maybe they have been successful in replacing science with religion in their science classes. If that is the case, then there is no chance that they will have been able to develop the science that would enable them to cross the immense void of space to get here just to do the same thing to us. But if they have been able to outgrow a concept that belongs in the infancy of any advanced civilisation, then perhaps there will, one day, be meaningful contact between them and us without the barbarity that has afflicted humankind throughout its history and is still with us “thanks” to religion.

On our little planet – this pale blue dot — history is quite clear: religion delivers intolerance, death and destruction; science delivers computers, the internet, medical wonders, the beginnings of space exploration – and the actual possibility of travel across interstellar space.

But if aliens really are on their way here,  we had better just pray hope that they don’t have religion.

## 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.