An article about the Law of Large Numbers in Psychology Today

#2
Hi Andy, nice to see you here again! I have transferred that article to my Kindle to read later, but I noticed your name appears as one of the references.

David
 
#4
This seems extremely unlikely...

Somebody actually married Andy?
Funny guy! Not only is the answer "yes", but she is the first and only person I've ever dated and we've been married since 1987. On top of that, I had a dream of her two years before we met. In the dream, I went through a series of events that, in the dream, I understood occurred about five years after we were already married. After waking from the dream, I was pretty surprised to have a memory of having been married for many years, considering I still hadn't dated anyone. Two years later, I saw my future wife in the doorway of a classroom and immediately recognized her as my wife from the dream. I didn't believe in precognition at the time, was an atheist, and thought that such subjects were a pile of poppycock, but had to admit, she looked exactly like the woman I was married to in the dream. About five years after we married, the scenes from the dream actually took place.
 
#7
Hi Andy!

I agree with what the article says, but it is missing a key contextual element; to wit, that you called out what was going to happen with the dice roll series before it happened.

So the article (you, actually) calculates the odds of rolling that series correctly and then goes into an appropriate discussion of why we can't relegate the experience to chance because we don't know if the number of people in the world having that dice roll series is greater than what we'd expect by pure random chance. However, once you called what the rolls were going to be immediately before they happened, we are no longer dealing with randomness. That whole model goes out the window, IMO; albeit it in a nuanced way.

What you were doing was forecasting. There is a whole set of statistical tools that are applied to analyzing forecasts - and I haven't had enough coffee yet to get into those at the moment :), but time is a factor. Also, the analysis should, IMO, be based on your trials alone. So every time you get the sense that you know what the roll will be, you record it and then roll. That is different than pure relative frequency of events over time or many trials (AKA, standard probability) with which the law of large numbers is associated.
 
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#8
H

What you were doing was forecasting. There is a whole set of statistical tools that are applied to analyzing forecasts - and I haven't had enough coffee yet to get into those at the moment :), but time is a factor. Also, the analysis should, IMO, be based on your trials alone. So every time you get the sense that you know what the roll will be, you record it and then roll. That is different than pure relative frequency of events over time or many trials (AKA, standard probability) with which the law of large numbers is associated.
Hello Eric! I'd love to read more about this, should you feel like putting it into the official record.

Best regards,

AP
 
#10
What you were doing was forecasting. There is a whole set of statistical tools that are applied to analyzing forecasts - and I haven't had enough coffee yet to get into those at the moment :), but time is a factor. Also, the analysis should, IMO, be based on your trials alone. So every time you get the sense that you know what the roll will be, you record it and then roll. That is different than pure relative frequency of events over time or many trials (AKA, standard probability) with which the law of large numbers is associated.
The Law of Large Numbers does not apply ex ante, nor in any other case where there is not a large number domain to sample from in the first place.
 
#13
I'm glad you understood what he wrote because I didn't. Looking forward to an explanation from someone. :)
Do you mean me with 'he'?

The Law of Large Numbers does not apply, if there is not a large number to begin with. That was the essence of Beitman's contention was it not?

Make no mistake: There is truth to the Law of Very Large Numbers, but it can only be properly applied when we have data for those large numbers.
 
#14
Do you mean me with 'he'?

The Law of Large Numbers does not apply, if there is not a large number to begin with. That was the essence of Beitman's contention was it not?

Make no mistake: There is truth to the Law of Very Large Numbers, but it can only be properly applied when we have data for those large numbers.
It may be the way you are stating this. Do you mean that because there are not a large number of events to compare it to, it cannot be compared? An irony of this is what isn't mentioned in the article. This is probably the most impressive example from the point of view of calculating the probability, and the probability being larger than other examples, but I have other similar examples. One happened within weeks of this event and followed a similar pattern. I was playing Scrabble with my wife when I suddenly had the inspiration to say "I will now withdraw all of the remaining "E" tiles from the bag (without looking, of course.) We had just started, and a quick look at the board showed there were eleven "E"'s remaining. I proceeded to pull out ten in a row before faltering and announcing I wouldn't get the last one, which I didn't. Another that happened when I was a teenager working at Comics & Fantasies comic book store in San Jose back in 1978 or so, I correctly announced every dice roll in advance of rolling it, but first determined the best roll for me and the worst roll for my opponent, my fellow clerk, an older teenager named Greg. Greg was so annoyed with my "luck" that he kept expressing his outrage during the game. At a certain point, I had almost all of his pieces on the bar, a nearly impossible position, as anyone who plays backgammon would know. Before I made the roll that would put his last couple of pieces on the bar, he picked up the board and tossed it into our front window.

The funny thing is that at the time, though I found the event remarkable, I did not think of it as paranormal or psi. To me, it was an uncanny string of amazing luck. What I actually thought at the time was that it was an example of the Law of Large Numbers, where improbable events are bound to happen to someone, somewhere, and I happened to be the someone that time. As I remember it, we were about forty rolls into the game, so the improbability of this is likely higher than the event described in the PT article, but it is much older and I don't recall exactly, so I don't like using this example as evidence. The most interesting event of this type to me, cannot have a probability calculation applied to it easily because the elements don't lend themselves to that type of analysis.

Here is the account, from my book:

"
Rhoda Gubernick of The Atlantic Monthly magazine hired me in the first week of February, 1990 to make a double portrait of the musicians John Cale and Lou Reed. A week later, the art was done and I shipped it to her in Boston by Airborne Express. The next day, she called at about four-thirty in the afternoon with an emergency. “Andrew, we’re just about to ship this out to Wisconsin to have it printed, but we noticed an error on your art.” She said that because I used color overlays, she didn’t notice at first that the color of the singers' skin in the portrait had the wrong percentage of black. “It’s a hundred percent right now, it will block out all the line art underneath it.”
It was a serious mistake and I wasn’t proud to have made it. Without having the art to compare it to, I could only guess, so I said to change it to seventy percent, assuming it should be dark.
After hanging up, I worried that the problem hadn't been fixed. The darkness of a color doesn’t matter as long as it contrasts with the line art. In this case, I made the line art lighter than usual. That meant that the new value I gave Rhoda might be just as bad as the first one. It took ten minutes to figure this out, leaving plenty of time to call Rhoda and give her another value. Doing so would require admitting that I’d made another mistake, but I didn’t want to do that.
This mistake was embarrassing. I wanted to call, but at the same time I really wanted to fix the art some other way. A phone call on its own wouldn't be good enough. I had to see the art to fix the error. This was before computers were used to ship art, and I didn't have a copy of the illustration at home. While Kitty sat in the kitchen eating dinner, I paced in the living room.
“What I really want,” I said, loud enough for her to hear, “is for the package to get waylaid on its way to the printers in Wisconsin and mysteriously arrive here without anyone knowing about it. I’ll fix it and send it on its way and everything will be fine.” The concept made no sense, but it was the only option I could think of.
I said this over and over again to Kitty as if it was a mantra. “Either I call, or it just appears, that’s it. It can’t happen any other way.” I tried to think of other options, but the only idea that would work was the one that was impossible. “It has to show up here tomorrow.” After two hours of pacing and repeating this, I went to bed.
The house phone woke me the next morning. There was a package waiting for me downstairs - I had no idea what it could be. The Airborne Express truck sat in neutral by the curb, punching clouds of vapor into the cold February air. The deliveryman handed me a box and a signature sheet. I was about to sign when I saw that the box didn’t have an air bill. I turned it over to check the other side, but it wasn’t there either. The Airborne employee was impatient, but I didn’t want to accept a package if it didn’t belong to me.
By now I was wide-awake and knew I wasn’t expecting anything. This box had no address information on it at all. It didn’t have a sender’s address, a recipient’s address, or even a single name to identify it in any way. It also looked like it had been opened and then artlessly resealed. To say it was suspicious was an understatement. “Where’s the air bill?” I asked.
“Dunno.”
“How do you know this is for me then?”
“My supervisor said to deliver it to this address.”
“Where is it from?”
“Dunno.”
“Do you mind if I look inside, just to make sure it’s for me?”
“It’s your box, man. Do whatever you like.”
I borrowed a letter opener from the doorman and sliced open the box. Inside was every article, every original photo, every original slide, and every piece of original illustration art for the entire issue of The Atlantic Monthly to which I had contributed. Buried in the middle, I found my illustration.
I signed for the box and ran upstairs. “Kitty!” I yelled when I got inside. “You are not going to believe this!” I pulled out my illustration and brought it over to my worktable. I found the color I needed to fix quickly, because Rhoda had taped over the original value with a new piece of tape and a note in her handwriting. I didn’t want any mix-ups, so I pulled her tape free of the art. Underneath, I found my original number. It wasn't written very clearly, but I was familiar enough with my own handwriting to see that it said “10%.” Rhoda had read the percent sign as part of the number. I hadn’t made a mistake at all, but if I hadn’t caught the later error that Rhoda and I had made jointly to “fix” the first, my first job for the Atlantic Monthly would have been a horrible misprint.
I called up Rhoda, “It’s Andrew Paquette. You’ll never believe what I’ve got.”
“Pardon me?”
“The package you sent to the printers yesterday arrived at my apartment in New Jersey this morning.”
“Oh my God.”
“Just tell me where it needs to go. I’ll send it on its way.”
I asked Rhoda if she knew of any past occasion when all the raw materials of an issue had been lost. “Never,” she replied, “Not in eighty years of publishing.”"
 
#15
It may be the way you are stating this. Do you mean that because there are not a large number of events to compare it to, it cannot be compared?
Great response...

Yes, ;;/? and two other contexts of the Law of Large Numbers Fallacy. There would be indeed three contexts of Law of Large Number Fallacies: Wrong context (your example story), inestimable (what you cite above here) and single event.

Wrong Context/Species

"What I actually thought at the time was that it was an example of the Law of Large Numbers, where improbable events are bound to happen to someone, somewhere, and I happened to be the someone that time."​
You had incorrectly assumed that a life should contain a certain number of paranormal-like events, when each one has to be taken in the contexts of its own species' set of probability, and not in the context of a 'life'. With some probability violations, a life may not be long enough to constitute a 'Large Number' domain. The context is actually the probability arrival distribution inside that species of odd event - and not the probability of an event inside a 'life'.​
If you predict that aliens will land on the White House lawn on Tuesday and it comes to pass - that event should be examined for incredulity as to its species of improbability - and not be excused as "every life has its oddities..." - which would suffer an ad hoc fallacy.​
Inestimable (Beitman's Point and Yours)

Abiogenesis proponents cite that, given so many planets - by the Law of Large Numbers, life had to appear somewhere, so why not our planet? Therefore abiogenesis is the null hypothesis.​
This is called and Appeal to Plenitude - and is really simply an ad hoc plea and not a real scientific explanation. As you mention above, this is a case where "there are not large number of events to compare our event to", so the LoLN does not actually apply.​
Single Event (ex ante)

When I was a young executive in Washington D.C. - my spouse and I were hanging around the townhouse one Saturday, when the phone rang. This was before we had Caller ID. It popped into my mind that this was a young lady with whom we had not spoken in 4 years (since her wedding), and that she was calling to explain that she was now expecting their first child. So I answered the phone "Hello Beth, I am so glad that you are expecting your first child." Dead silence on the other end of the line...​
To my amazement it was Beth, and she was calling to announce they were expecting their first child.​
Now - I had never tried a stunt like that before, and never tried such a thing again. It was a single ex ante event. The Law of Large Numbers does not apply in this case.​
Note: if one predicts that a meteor will cross the night sky in the next 30 seconds, that is not an example of ex ante. That is an example a priori reasoning.
 
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#16
Great response...

Yes, ;;/? and two other contexts of the Law of Large Numbers Fallacy. There would be indeed three contexts of Law of Large Number Fallacies: Wrong context (your example story), inestimable (what you cite above here) and single event.

Wrong Context/Species

"What I actually thought at the time was that it was an example of the Law of Large Numbers, where improbable events are bound to happen to someone, somewhere, and I happened to be the someone that time."​
You had incorrectly assumed that a life should contain a certain number of paranormal-like events, when each one has to be taken in the contexts of its own species' set of probability, and not in the context of a 'life'. With some probability violations, a life may not be long enough to constitute a 'Large Number' domain. The context is actually the probability arrival distribution inside that species of odd event - and not the probability of an event inside a 'life'.​
If you predict that aliens will land on the White House lawn on Tuesday and it comes to pass - that event should be examined for incredulity as to its species of improbability - and not be excused as "every life has its oddities..." - which would suffer an ad hoc fallacy.​
Inestimable (Beitman's Point and Yours)

Abiogenesis proponents cite that, given so many planets - by the Law of Large Numbers, life had to appear somewhere, so why not our planet? Therefore abiogenesis is the null hypothesis.​
This is called and Appeal to Plenitude - and is really simply an ad hoc plea and not a real scientific explanation. As you mention above, this is a case where "there are not large number of events to compare our event to", so the LoLN does not actually apply.​
Single Event (ex ante)

When I was a young executive in Washington D.C. - my spouse and I were hanging around the townhouse one Saturday, when the phone rang. This was before we had Caller ID. It popped into my mind that this was a young lady with whom we had not spoken in 4 years (since her wedding), and that she was calling to explain that she was now expecting their first child. So I answered the phone "Hello Beth, I am so glad that you are expecting your first child." Dead silence on the other end of the line...​
To my amazement it was Beth, and she was calling to announce they were expecting their first child.​
Now - I had never tried a stunt like that before, and never tried such a thing again. It was a single ex ante event. The Law of Large Numbers does not apply in this case.​
Note: if one predicts that a meteor will cross the night sky in the next 30 seconds, that is not an example of ex ante. That is an example a priori reasoning.
Based on your answer, you may enjoy my first published journal article on the subject:
https://www.scientificexploration.org/docs/26/jse_26_3_Paquette.pdf

You may also enjoy one of my later articles, written with some help from Daryl Bem:
https://www.researchgate.net/public..._Dream_Journal_Comprising_Nearly_12000_Dreams

I wrote the first one just as I was beginning my PhD studies at King's College, London (now completed). The second article was written a bit later, after I'd gotten better at writing articles like these. My favorite article, also the most recent, was rejected by the JSE because they thought I was "evangelizing". I disagreed with their assessment, and made some effort to avoid even the appearance of promoting a religious perspective but I did want to discuss the possibility that the connection between genuine psi and religious beliefs suggests the possibility that at least some of those beliefs are correct. That article got published in the International Journal for Dream Research:
https://www.researchgate.net/public...endent_agents_and_spiritual_content_in_dreams

Best regards,

AP
 
#17
I didn't believe in precognition at the time, was an atheist, and thought that such subjects were a pile of poppycock,
The funny thing is that at the time, though I found the event remarkable, I did not think of it as paranormal or psi.
Hi Andy

Was there any particular event that changed your worldview or was it a gradual thing? I’m interested in this. If so, have you written about it?

Do you continue to experience events like those described in Dreamer? What do you think about some people having them while the majority don’t, at least not to the point where they are very persuasive of something unusual.
 
#18
I'm glad you understood what he wrote because I didn't. Looking forward to an explanation from someone. :)
Hi Andy,
I thought the ethical skeptic was basically reiterating what was said in the article, "The existence of 7 billion people on our planet is only relevant to our coincidence experiences if we know how many of those 7 billion people have or have not experienced coincidences as striking as the ones we’re considering. That is, large numbers are only relevant if we have data for those large numbers."

What is interesting to me from the statistical analysis standpoint is that, in addition to the forecasting aspect, Kitty's roll results and your roll results were not really independent. Your forecast mandated that you each sequentially obtain a certain roll in order. Thinking about this....
 
#19
I think that my interest in bad science in general, was the realisation that a lot of modern science seems to be powered on false reasoning of various sorts. Using the large numbers argument without making any attempt to estimate the frequency of the events you are trying to explain, is just one example.

David
 
#20
I think that my interest in bad science in general, was the realisation that a lot of modern science seems to be powered on false reasoning of various sorts. Using the large numbers argument without making any attempt to estimate the frequency of the events you are trying to explain, is just one example.

David
David,
Yes. Would a billion monkeys banging away on a keyboard for a billion years ever, by chance, create an exact copy of 'War and Peace'? Would they even create an exact copy of the text of the See Spot Run children's intro to reading book? I think not.

Would they eventually type out each word in either book? Yes, but not strung together in the right order to convey meaning; i.e. sentences and paragraphs.

The probability of obtaining the exact order of words in sentences and paragraphs, by random chance, is infinitesimal (but would even an infinite number of monkeys typing into infinity ever get it right?). Obviously, probability calculation based on randomness breaks down if there is intelligent design behind the outcome.

Statistical analysis has no way of telling us if there is intelligent design behind an event's occurrence. It is up to the analysist to decide whether or not the probability is so small that an event is not a mere coincidence/law of large numbers. What I like about working in the insurance business world is that we can observe unusual events or patterns and decide that it's not random chance and take action - and we're usually right.

We flip the skeptical science approach on its head. If it's too low a probability to be random chance, then it's not random chance, though I note that is how science itself looks at most things. Law of large numbers only gets applied when its the paranormal hypothesis, or some other out of vogue concept, being asserted. If it's a crumby drug trial with a sample of 90 and you get a P <.05, then you've created a miracle cure that can go to market for big profits. No need to replicate multiple times to ensure that other samples also come in at the same or smaller Ps.
 
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