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.

MC900439329A 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.

calculate winningsDouble 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.


4 responses to “Statistically Impossible? Probably Not

  1. Claiming expertise in something is plain arrogant!


  2. Maponos,

    You appear to be claiming expertise in arrogance. There’s a certain irony there.

    You are welcome to comment here if you can contribute something meaningful to the subject, but abuse is not constructive. If you are a believer in the paranormal, you are not a very good representative for your side of the argument.

    No one will be prevented from commenting here on the grounds that they disagree with me, but oafish comments like the one you have just made will probably not get past moderation.


  3. Good post; well-written and a breath of fresh air on the topic of several amazingly common misconceptions.

    However, I do have one small quibble, regarding the passage: “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.”
    It isn’t actually a statical certainty, just an extremely likely event. Theoretically, everyone who wins a lottery jackpot could just never buy another ticket, for example.

    But that problem is only one word; I still agree with the general idea of that part, and also with every other point you make.


    • Michael,

      Thanks for your comment.

      I won’t quibble with your quibble, except to say that coincidences are expected. As they say, the biggest possible coincidence would be the absence of coincidences.

      Now, if psychics could predict these coincidences before they happened rather than rationalising them afterwards…


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