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law of truly large numbers (coincidence)

"That a particular specified event or coincidence will occur is very unlikely. That some astonishing unspecified events will occur is certain. That is why remarkable coincidences are noted in hindsight, not predicted with foresight."--David G. Myers

The law of truly large numbers says that with a large enough sample many odd coincidences are likely to happen.

For example, you might be in awe of the person who won the lottery twice, thinking that the odds of anyone winning twice are astronomical. The New York Times ran a story about a woman who won the New Jersey lottery twice, calling her chances "1 in 17 trillion." However, statisticians Stephen Samuels and George McCabe of Purdue University calculated the odds of someone winning the lottery twice to be something like1 in 30 for a four month period and better than even odds over a seven year period. Why? Because players don't buy one ticket for each of two lotteries, they buy multiple tickets every week (Diaconis and Mosteller).

Some people find it surprising that there are more than 16 million others on the planet who share their birthday. At a typical football game with 50,000 fans, most fans are likely to share their birthday with about 135 others in attendance. (The notable exception will be those born on February 29. There will only be about 34 fans born on that day.)

You may find it even more astounding that "In a random selection of twenty-three persons there is a 50 percent chance that at least two of them celebrate the same birthdate" (Martin).

On the other hand, you might say that the odds of something happening are a million to one. Such odds might strike you as being so large as to rule out chance or coincidence. However, with over 6 billion people on earth, a million to one shot will occur frequently. Say the odds are a million to one that when a person has a dream of an airplane crash, there is an airplane crash the next day. With 6 billion people having an average of 250 dream themes each per night (Hines, 50, though I don't think I've ever had more than 5 or 6 dream themes a night), there should be about 30,000 to 1.5 million people a day who have dreams that seem clairvoyant. The number is actually likely to be larger, since we tend to dream about things that legitimately concern or worry us, and the data of dreams is usually vague or ambiguous, allowing a wide range of events to count as fulfilling our dreams.

Carl Jung, like many people who have experienced an uncanny pairing of events, did not think such happenings are mere coincidence. He developed  the notion of synchronicity to explain "meaningful coincidences." He described synchronicity as an acausal principle that links events having a similar meaning by their coincidence in time. However, if you think of all the pairs of things that can happen in a person's lifetime and add to that our very versatile ability of finding meaningful connections between things, it then seems likely that most of us will experience many meaningful coincidences. The coincidences are predictable and we are the ones who give them meaning. Given the fact that there are billions of people and the possible number of meaningful coincidences is millions of billions, it is inevitable that many people will experience some very weird and uncanny coincidences every day.

Finally, clusters of coincidences can seem designed or the result of a preordained pattern to someone who is very selective in his thinking, such as Uri Geller. After the anti-American terrorist attacks on September 11, 2001, Geller posted his thoughts on the number 11. He asked everyone to pray for eleven seconds for those in need. Why? He was convinced that there was a cryptic, numerological message in the events that occurred that day. In fact, he admits that he has had a long-term relationship with the number 11. He thinks that 11 "represents a positive connection and a gateway to the mysteries of the universe and beyond." But here is what he had to say about the terrorist attack and how 11 relates to this day of infamy:

Geller's Mysterious Number 11

The date of the attack: 9/11 - 9 + 1 + 1 = 11

*September 11th is the 254th day of the year: 2+5+4 = 11

*After September 11th there are 111 days left to the end of the year.

*119 is the area code to Iraq/Iran. 1 + 1 + 9 = 11 (reverse the numbers and you have the date)
[not true: Iran's country code is 98 (9+8=17), Iraq's is 964 (9+6+4= 19); Saudi Arabia's code is 966 (9+6+6=21.)]

*Twin Towers - standing side by side, looks like
the number 11

*The first plane to hit the towers was Flight 11 by
*American Airlines or AA
- A=1st letter in alphabet so we have again 11:11

*State of New York - The 11th State added to the Union

*New York City - 11 Letters

*The USS Enterprise is in the gulf during the attack; its ship number is 65N 6+5=11 [Or, maybe not. I received the following in an email: "We were actually about 12 hours away from the equator (I know this because we were about to have our crossing the line ceremony that night) on our way to South Africa. We had left the Gulf about a week before that and were relieved by the USS John Stennis (CVN-74 but also adding up to 11)."]

*Afghanistan - 11 Letters

*The Pentagon - 11 Letters

*Ramzi Yousef - 11 Letters (convicted or orchestrating the attack on the WTC in 1993)

*Flight 11 - 92 on board - 9 + 2 = 11

*Flight 77 - 65 on board - 6 + 5 = 11

*The # of story is 110 (2x) 110 - 110 Remember that the zero "0" is not a number, so we have 11:11

*The house where they believed to have lived had # 10001 again don't count the zero.....

Names that have eleven letters
*Air Force One, *George W. Bush, *Bill Clinton, *Saudi Arabia, *ww terrorism, *Colin Powell U.S. Secretary of State

*Remembrance day is November 11.... 

is also the 11th month

*Mohamed Atta, the pilot that crashed into the World Trade Centre.

, American Airlines AA = 11

Geller also claimed that "there will be more information coming in" and urged readers to e-mail him with more findings of this kind. Geller also wrote: "I would encourage everyone to send out this message to all your family, friends or business acquaintances and try to put this in the right perspective."

Let me try to put this in the right perspective. 

Accumulating more findings like these coincidences between the number 11 and other things should be easy, since there are countless items that can be made to relate in some fashion to the number 11 (or 12, 13, or just about any other number or word). E.g. the country code for Pakistan is 92 (9+2=11), a Boeing 757 holds about 11,000 gallons of fuel and the wingspan is 155 feet (1+5+5=11), Nostradamus and Billy Graham have 11 letters, any word with 8 letters just put 'the' in front of it; any word with 9 letters, just put 'ww' or any two-letter word in front of it, any word with 10 letters, just put 'a' in front of it or add an s to the end of it, etc. It is especially easy to do this since there is no specific guide before we begin our hunt for amazing facts as to what will and what won't count as being relevant. We have a nearly boundless array of items that could count as hits. Unfortunately for Geller and others who are impressed by these hits, there is also an even larger array of items that could count as misses. Geller doesn't see them because he isn't looking for them.

If we start hunting for items that seem relevant but don't fit the pattern, we will soon see that there is nothing special about Geller's list or the number 11. Only by focusing on anything that we can fit to our belief and ignoring everything that doesn't fit (confirmation bias) can we make these coincidences seem meaningful.

Carroll's List of Meaningless Coincidences

  • the planes hit a little before and a little after 9 am

  • there were 4 flights that crashed with 266 people in them; 2+6+6= 14

  • one plane was a 767 (7+6+7 = 20)

  • the other a 757 (7+5+7 = 19)

  • the 767 has 20,000-gal fuel capacity

  • the 757 has a 124-foot wingspan (1+2+4 = 7)

  • a 767 has a 156-foot wingspan; 1+5+6 = 12

  • one tower was 1,362 feet high; 1+3+6+2 = 12

  • the other was 1,368; 1+3+6+8 = 18

  • the supports are 39 inches apart; 3+9 = 12

  • the buildings were leased for 3.25 billion; 3+2+5 = 10  

  • the other flights were AA 77  (7+7 = 14) and UA 93 (9+3= 12) and UA 175 (1+7+5= 13)

  • flight 11 had 81 passengers (8+1 = 9)

  • flight 77 had 58 passengers (8+5 = 13), 4 crew members and 2 pilots (total=64)

  • Boston has 6 letters

  • Massachusetts has 13 letters

  • Pennsylvania has 12 letters

  • Washington D.C. has 12 letters

  • Los Angeles has 10 letters

  • The number of hijackers was 19 (9+1=10)

  • Names and words that don't have eleven letters in them: Osama bin Laden, Khalid Al-Midhar, Majed Moqed, Nawaq Alhamzi, Salem Alhamzi, Satam Al Suqami, Waleed M. Alshehri, Wail Alshehri, Abdulaziz Alomari, Dick Cheney, Laura Bush, Iraq, Iran, Pakistan, Muslim, Islam, Pentagon, World Trade Center, terrorism, jihad, Taliban, Koran, United States, anti-American, murder, fire, hell, stupid.

What Geller and other numerologists are doing is a game, a game played with numbers and with people's minds. Sometimes it is amusing. Sometimes it is pathetic.

See also confirmation bias, dream, Littlewood's law, magical thinking, numerology, selective thinking, and synchronicity.

reader comments

further reading

books and articles

Diaconis, Persi and Frederick Mosteller, "Coincidences," in The Encyclopedia of the Paranormal, ed. G. Stein (Amherst, N.Y.: Prometheus Books, 1996).

Hines, Terence. Pseudoscience and the Paranormal (Buffalo, NY: Prometheus Books, 1990).

Coincidences: Remarkable or Random? by Bruce Martin in The Skeptical Inquirer, September/October 1998.

Paulos, John Allen. A Mathematician Reads the Newspaper (Anchor Books, 1996).

Paulos, John Allen. Innumeracy: Mathematical Illiteracy and Its Consequences (Vintage Books, 1990).

Strogatz, Steven H. (2004). SYNC: How Order Emerges From Chaos In the Universe, Nature, and Daily Life. Hyperion.


Coincidences: the truth is out there Teaching Statistics by Robert Matthews

A Brief Treatise on Coincidence by Michael A. Thalbourne

The Power of Coincidence: Some Notes on "Psychic" Predictions (2000) by Robert Novella

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