William M. Briggs

Statistician to the Stars!

Can You Read My Mind?

Ghosts, ESP, telekinesis, astrology, and other assorted oddities are back in view. One of the “SyFy” channel’s most popular series is a show about hunting apparitions. The movie Men Who Stare at Goats will soon be upon us. It is important to understand that people believe in the paranormal not entirely because of wishful thinking. Many have conducted rigorous tests and have, through misuse or misunderstanding of statistics, come to think they are on to something. So I wrote a book to show people how mistakes can be made and how easy it is to fool oneself into accepting bad evidence. Here is an excerpt from Chapter 3—a test of telepathy—which should give you a good idea of the tone. I’d be very interested to hear people’s results of trying it.

Telepathy The ability to gather information about someone else’s thoughts through non-verbal, non-sensory means. Sometimes known as mind-reading or extra-sensory perception (ESP).

Quick—I’m thinking of a number between one and five! Can you read my mind? Was your guess three? That’s the most common guess. And if your guess matched my number maybe you have telepathy. Or maybe not. To convince me you had extraordinary powers, it would depend on how difficult, and how surprising, your guess was. If my number was three your hit is not necessarily that surprising because you could have guessed correctly by luck.

The number I was thinking of was e. That’s equal to about 2.71828. What? You’ve never heard of this weird number e1 and you think I was cheating? Well, I never said my number had to be an integer, that is, a whole number like one, two, three, four, or five. I specified a number between one and five and e=2.71828 is certainly in that range.

So You Think You're Psychic

But maybe you did guess correctly; even so, I wouldn’t be terribly surprised. Why? Because you might have used the fact that I was mathematically minded, guessed I was being tricky, and figured I would use an extremely common number, a number mathematicians encounter daily. Another number might have been π (which is about 3.14159). There are lots of mathematically common numbers between one and five, and there are an uncountably infinite selection of other numbers, which it is why it is necessary to set up an experiment, in advance, that allows me to quantify the chances of you and I thinking of the same one merely by luck. Controlled experiment are needed in which the only thing that is left to chance is the guessing (or mind reading) itself.

The end of this experiment, where you guessed my number between one and five, is the first example of a multiple endpoint; multiple endpoints were described in Chapter 2. The test, first announced with a “Quick!”, was not completely defined because I never specified the set of possible answers you could choose from. You were free to make any guess you wanted, whether an integer or some strange number and then, after the fact, when I revealed my number, transform your score to something that seemed surprising by arbitrarily defining your choice of possible answers. For example, you first might have thought that the possible answer were the numbers one through five. If you were to say the set of answers included all real numbers, the chance of you guessing my exact number by chance would be exactly 0 (Chapter 2 explains why this is true). That’s as surprising as you can get and terrific evidence of psi. On the other hand, I could retort that I limited my choice to e and π, which means a match happens half the time, thus a correct guess is not in the least surprising. This is why only controlled experiments with fixed endpoints are convincing.

Of all paranormal abilities it is probably telepathy (sometimes given the important sounding academic title anomalous information transfer) that most excites the imagination, engenders the highest interest, and generates the most enthusiasm. Almost everyone feels they have had some personal experience with various forms of telepathy, whether it’s picking up the phone just as someone calls, or thinking the same thought at the same moment as the person you are with. It is the ability that, even if you suspect other psi claims to be false, you are sure to think there is some truth to this one. Parapsychologists think so too and more experiments have been done in this area than any other.

Casual instances of telepathy are common. Imagine you and your sister are together, chatting over coffee about the hair dresser, about the bad hair style your mutual friend Edith effects to wear in public and so on, and then you have a flash, “Say, Judy, I was just thinking of that time when Bob chopped off half his moustache.” Judy says, “Me too! Isn’t that extraordinary! I must have been reading your mind!”

Not necessarily.

It may be the case that the more time you spend with someone the more it’s likely you will begin to think alike (maybe even look alike) and share the same thoughts. The more time together the higher the chances become that, if you want to think biologically, your brain’s neurons fire along identical pathways, using as input similar circumstances and shared sensory experiences, your minds arriving at nearly identical conclusions. Your common history and education, mutual background, and communal social activities help you interpret the world around you in a similar manner. It would be more surprising for two people like this—a married couple, for example—not to be thinking of the same things at the same times for a lot of their life. So how surprising is it that you and someone you know very well both happen to be thinking of the same thing? And how can you quantify the correspondence of thoughts? These tests can help answer that question.

Test Number One: The Card Test

This is a very easy and clean test. It is also very traditional in that the first formal tests of telepathy were very much like this. The materials needed are a deck of cards and your notebook. The set up is simple and quick. Cards are used because most people have at least passing familiarity with their shapes and values: this acquaintance is thought to ease mental transmission. The only difficulty may be in securing the help of a friend—but who doesn’t want to learn if they are telepathic? The next time you have a party you can assemble people into groups of two and run the test concurrently for each group.

WHAT YOU WILL NEED

  1. A deck of new clean playing cards (poker or bridge), or a deck of Tarot cards.
  2. At least one friend.
  3. Your notebook with two ruled columns (these can be done by hand). One column is headed Card and the next Guess.
  4. (Optional) A watch with a second hand.

WHAT YOU WILL DO

Details will be given for an ordinary deck of playing cards. Tarot cards will work but you must first remove the cards of the Greater Arcana (Death and all his brothers) and the Knaves of each suit (pentacles, swords, etc). Only 52 cards can remain in the deck for the probability score to be valid.

  1. Be sure to first remove all jokers from the deck of playing cards. Only 52 cards should remain.
  2. Riffle shuffle the cards at least seven times to insure they are thoroughly and randomly mixed. Riffle shuffling is the type where you take approximately half the cards in each hand, held lengthwise, and both piles are flipped towards each other to mix them. Square the deck when this is done. Straighten them out so they are all in one neat pack. For those who are interested, it was mathematically proven that cards should be shuffled at least seven times to attain true randomness (meaning the order is essentially unpredictable to you), so do not be lazy here.
  3. Decide who will be the sender and who will be the receiver. You can certainly swap roles in subsequent tests.
  4. The sender picks up the first card from the deck and notes it down in the Card column of the notebook. Shorthand should be used. Assume the first card was the Queen of Spades: QS would be written. Use ‘D’ for diamond, ‘C’ for clubs, ‘H’ for hearts, and ‘S’ for spades.
  5. The sender then concentrates on the card for a prescribed amount of time (say 30 seconds) during or immediately after which the receiver states their impression of the card.
  6. The sender writes this impression in the Guess column next to the Card column.
  7. Card number two is selected and the test repeated, and so on.
  8. After all cards are expended the sender then tallies the results by circling the matching results. The number correct is compared with the Telepathy Card Scoring Table for the score.


Telepathy card scoring table. This table shows the probability of getting n or greater correct matching cards from a deck of 52. Only correct guesses up to 6 are indicated, as it would be highly unlikely to get 7 or more correct. As it is, there is strong telepathic evidence by getting only 4 or more matches—the probability of this happening by chance is only 0.019 (this means it would happen by chance about 2 times for every 100 trials).

Probability of Getting at least n Correct Cards
n 0 1 2 3 4 5 6
p 1.000 0.640 0.260 0.080 0.018 0.0032 0.0005

This is only one possible scoring table because the guessing strategy used by the receiver can modify the probability results. A quick example will show how. Imagine the receiver always said Three of Clubs for each guess. Then she must get at least one guess correct, this being the time when the card was the Three of Clubs. All other cards will be wrong (of course). The makes the probability of getting 1 or more correct matches 1, and forces the probability of getting 2 or 3 etc. or more correct matches to equal 0. This table assumes the receiver is guessing freely, each time making a selection from any of the 52 possible cards, and the receiver may guess the same card more than once if she likes.

WHAT COULD GO WRONG

As long as both parties are careful about card handling (no peeking etc.), little can go wrong on this test except for the problem of feedback. Feedback is where the sender, consciously or not, informs the receiver if her last guess was correct. Perhaps the sender subtly smiles for correct guesses and frowns or grimaces for wrong ones. You might think this is not a big deal. Let me give you a simple example of why it is an enormous big deal.

Imagine a deck with only three cards, a Jack (J), Queen (Q), and King (K) (the suits are not important here). I shuffle the cards well and then select the top one, then try to mentally transmit its image to you. You say J. What is the probability you are correct? Well, there is one J and three cards, so the chance is one in three (p=1/3). I say nothing about whether your guess was correct or not. On to the next card. Again, I pick it up and concentrate. You say, for example, K. What is the probability this one is correct? Same thing, one in three. Finally, we do the last card. You pick up a strange vibration, reconsider your first choice, and say J again for the last guess (the point here is that you can say any of the three cards because you don’t know which cards are still in the deck). The No Feedback Scoring Table lays out the possible scores.


No feedback scoring table for a deck of three cards.

Probability of Getting at least n Correct Cards
n 0 1 2 3
p 1.000 0.700 0.260 0.037

Now let’s try the test again, this time with feedback. The cards are shuffled and I concentrate on the first one. You say J. “That was a hit,” I announce (with a one in three chance of happening, as before). I tell you it was correct. Now the next card. You say Q. What is the probability that you are right? It’s no longer 1/3 because you know the J is not one of the remaining two cards. You know, from feedback, that only the Q and K are left. So the probability of being correct by chance went from 1/3 to 1/2—a pretty big jump.

Let’s say your guess was right again, and the second card was a Q. I tell you this. This leaves one card on the table. What is the probability your last guess will be right? Well, if you have been paying attention, it is 1! You know it is the K without having to guess. Therefore, the probability of a correct guess has grown from 1/3 without feedback to 1 in the case of full feedback. This is an enormous increase!

The Feedback Scoring Table, given below, also depends on the guessing strategy used by the receiver. This is the one that is generated by an optimal guessing strategy, that is, one that uses all the feedback in the most efficient manner so as to maximize the probability of getting three correct guesses.

You can quickly see that the probability, without feedback, of getting all three cards correct went from 0.037 to 0.20, a whopping increase! In other words, if you had got all three correct without feedback it appeared you had psychic powers (based on the low probability score). But getting all three right with feedback is a completely unsurprising result and is not convincing evidence of psychic functioning.

Another thing to note is that, with feedback, you are guaranteed to get at least one card right, while there is only a 70% chance of this without feedback. It is also possible to examine the average number of hits you would get if only chance were operating. For this experiment, without feedback, you would expect about 1 hit. With feedback this doubles to 2 hits.


Feedback scoring table. The table to use with a deck of three cards when feedback is given. This table represents the optimal guessing strategy using full feedback.

Probability of Getting at least n Correct Cards
n 0 1 2 3
p 1.00 1.00 0.60 0.20

It becomes complicated to account for feedback when dealing with a full deck of 52 cards. It can been done, however, and a statistical test has been developed to account for it. This test is too complicated to include in this book but interested readers can look up the reference in Chapter 15.

The point to remember is that, with feedback the probability of getting a certain number of hits is higher than without, thus making the possibility of telepathy less likely. If feedback is suspected the tests must be modified and, in general, this is a very difficult thing to do. The simplest solution is to eliminate feedback.

One potential way to eliminate any possibility of feedback is to separate the receiver and sender. Stationed in different rooms each could agree to concentrate on one card for every minute. As long as everyone’s watches are synchronized, this should work quite well. (See the Ganzfeld test for some tips.)

A more insidious form of feedback is pencil reading. A well-known magician’s trick is to watch the pencil of a volunteer as he writes down the card to be transmitted. The magician makes his guess by watching the pencil as the volunteer writes. You would be surprised at how easy this is to do. Keep your notebook and pencil hidden. This type of cheating has been used in professional telepathy tests too, and the hapless researchers in these experiments found themselves in the position of reporting spurious positive results. Remember our golden rule: eliminate all sensory feedback!

Another form of naughtiness that is often found is misinterpreting or reworking the results after the experiment is over (multiple endpoints again). For example, suppose that you go through this test and get two hits, which is not very indicative of telepathy. But you notice that in five instances every time you guessed a card it came up on the next draw instead of the current one. That is, it appeared you were “seeing ahead” into the deck. You would have got five hits had you counted card-ahead guesses which now looks like great evidence of psychic ability. Wrong! The experiment was not to count how many cards you guessed right in the future but how many current ones you got right.

Here is a specific example of how this post-experiment data mining is wrong. Return to the pack of only three cards, the Jack, Queen, and King. Your goal is to guess the card names as before. A glance at the No Feedback Scoring Table, whose results still hold, confirms that the probability of getting all three guesses wrong is 0.30 (this is calculated using the principle that the probability of getting all wrong is one minus the probability of getting at least one right). Likewise, the chance of getting at least one correct is 0.70. Imagine the order of the cards and your guesses was the following:

Card Order Your Guesses
J Q
Q K
K J

Notice that none of your guesses was correct. But all is not lost because you notice that, lo and behold, when you guessed Q, although it wasn’t a direct hit, it was only one card away from the one you guessed. Using that logic to rescore the test shows that you now have two hits (the K and Q were only one card away)2. Naively using the No Feedback Scoring Table gives a score of 0.26. This is wrong because you have redefined the experiment and this changes the probability structure of the score. It is the case that the probability of getting at least 2 correct guesses is 0.83, which is three times higher than before (incidentally, the probability of getting at least one becomes 1, meaning you will always get at least one hit).

For a full deck of 52 cards, the probabilities in the scoring table are modified in a similar, but mathematically complicated, fashion—all probabilities increase making it less convincing that you are demonstrating psychic ability.

There is another even bigger problem with the probability calculation just given. It is incomplete. This is because you saw that your guesses were one away, but what if you saw your card was three away? Or four? Or some other bizarre pattern? There is a limitless number of possible patterns that may account for your guessed card being at other places other than the current card. And there is no way to know in advance what they are. All that can be said is that they all modify the probability score towards chance and away from psychic ability. Since there is no way to know in advance the only logical thing to do is to disallow all results except the exact results the test specified. If you truly feel you’re always better at guessing cards one in advance, set up a new experiment (like the one outlined under precognition) and then interpret the results under that experiment. It is impossible to give probability score tables for all the clever ways people can reinterpret results. You might decide to look two ahead, three ahead, or some random number ahead. You may also decide to look ahead and behind. You may change things to look only at the color, suit, or value of the card. There is no way for me to out-guess the creativity of the reader. Therefore, it is imperative that you stick to the original test.

Unfortunately, many published results in prominent parapsychological journals fall into the trap of redefining the experiment after the fact. These researchers post-interpret the results looking for significance. Some, wiser than others, do not try to claim “statistical significance” when they do this, and instead claim the results are “intriguing,” but this circumlocution doesn’t change the bad result into a good one. Attaching or not attaching a significance number does not make the practice right. You cannot, under any circumstances, claim results are meaningful when you did not set out to test them. Again, if you feel you are obtaining results that are not associated with the experiment, set up a new experiment specifically to test for these results.

Of course, if you’re doing this test at a party or other informal get-together you cannot disallow the possibility of cheating after one too many pinot grigios it’s impossible to tell what one of your friends might do. It’s human nature to want to impress other people and the temptation may be too great to “bend” the results in the right direction. This could all be in fun, but don’t fool yourself. If you feel you have a friend who is particularly adept, schedule a time with her alone and do the test under rigorous conditions.

The last big thing that can go wrong, and often has, is bad or incomplete randomization. If you fail to shuffle the cards correctly, particularly between trials, you will seem to do better than chance, but this may only be because you have, inadvertently or not, memorized certain patterns in the cards. Shuffle well to avoid this.

[Copies of the book may be had here.]

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1The number e is a number mathematicians like me meet daily. It is the base of the natural logarithm.

2It might be argued that the J guess was only one card away too, if the set of cards is envisaged as a circular chain. Whether or not it makes sense to ask for this interpretation depends on how many cards you want to get right. If viewed as a chain it is then impossible to get any wrong guesses.

2 Comments

  1. 2 … no 3! Aaaaaaaaaaaaaaaaaaaaaaaaaaaaaarrrrrrrrrrrrrrrrrrrrrrrrrggggggggggggggggggggghhhhhhh!!!!

  2. In your JQK example with feedback, it seems to me that you have 1/3 chance of getting the first card right, and then (if successful) a 1/2 chance of getting the second right, and then (if successful) the third card is certainly right. So by my calculation the probability of all three correct is 1/6 = 0.17, rather than the probability of 0.20 that you’re giving.

    Am I missing something here?

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