Have you ever experienced a coincidence so unbelievable, you felt like calling it a 'miracle?'

Maybe you were humming a song to yourself, and then walked into a room where the exact same song was playing. 

Or maybe today was the day you said something out loud you'd been thinking about for a while, like, "I need to try something new"... and then the commercial on TV repeated those exact words back to you. Which is what happened to this girl.

British mathematician David J. Hand (below) has been studying this phenomenon, which some call "synchronicity," for years. Yet according to Hand, when we use statistics to determine the likelihood of these so-called 'miracles,' it turns out they are not just possible, they are actually likely to happen.


Hand calls this idea the 'Improbability Principle,' the notion that any unbelievable coincidence you can think of is bound to happen eventually, so long as there are enough opportunities for it to occur.  

In his book The Improbabilty Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day, Hand uses a series of thought experiments to illustrate this idea, beginning on the next page with what he calls the "Birthday Problem."  

David J. Hand uses the example of the Birthday Problem to illustrate how coincidences are much more likely than we tend to think they are.

Let's say you're a teacher in a modestly-sized classroom, hoping to find two students who have the same birthday. If you've never encountered this problem before, you might think this is a hopeless exercise with only one classroom full of students. There are 365 days in a year, so logically, you would need at least half that many students to find a birthday match between two of them, no?

Nope. 23 students. That's all you need to make it more likely than not that two of them will have exact same birthday. With 23 students in a room, there are actually 253 possible pairings between them. All we need is for one of those 253 pairs to share the same birthday. And there's a 51% chance that will be the case.

Suddenly, two out of 23 people having the same birthday doesn't sound so miraculous, does it? And if you don't believe me, you can try it yourself as an icebreaker game.

But wait. Does this principle still apply to something incredibly unlikely, like winning the lottery? Find out on the next page.

On September 6, 2009, a Bulgarian lottery host read six winning numbers from the lottery machine: 4, 15, 23, 24, 35, and 42. A day passed, and no one claimed the correct numbers.

Four days later, the next set of winning numbers were announced. The host was stunned as he read them aloud: 4, 15, 23, 24, 35, and 42. In the 52-year history of the lottery, the same lottery numbers had never repeated twice in a row. A statistician reported that the odds of this happening were 4.2 million to one. 

This time, an unprecedented 18 people had the winning numbers, and split the relatively low jackpot, each winning just 10,164 leva (around $2000 USD). A scandal ensued, and despite the low payout, Bulgaria's Sports Minister Svilen Neyko launched an investigation to determine if there had been foul play. But nope, the probe confirmed that the draw was legit. 

This is happening for the first time in the 52-year history of the lottery. We are absolutely stunned to see such a freak coincidence, but it did happen," said a lottery spokeswoman at the time.

According to Hand, we shouldn't be surprised at all. The Improbability Principle tells us that any specific event will happen eventually if there are enough opportunities for it to happen.

So, how many opportunities are there for the same lottery numbers to repeat? 

The short answer? A lot. Even just in the Bulgarian lottery, two draws occur each week, which equates to 104 draws in a year. Over the course of 52 years since the Bulgarian lottery began, that's 5,408 draws.

Then let's add in the law of combinations from the Birthday Problem. In that problem, we asked, how many possible pairs would we need to increase those odds to at least 50%? In a single draw, the chance of any particular set of 6 numbers coming up twice in a row is one in 13,983,816. But each time we increase the number of draws, the number of possible pairs increases exponentially.

4,404 draws. That's all we need. In that time span, it's more likely than not that the same numbers will be drawn twice. 

5,408 draws. That's how many it actually took. Which means, contrary to what the spokeswoman said, we shouldn't be shocked that this happened. If anything, it should have happened sooner. 

So there you have it. Most of what we might call everyday 'miracles' can be explained by probabilities and statistics.

Does that mean I won't be a little freaked out the next time a song plays that I've been humming all day, or a TV ad repeats the sentence I just spoke out loud? Probably not, but at least now I know what's really going on.

For more on how math explains strange coincidences, you can read this article by Hand himself, or you can check out his book.

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From Pride and Prejudice to The Handmaid's Tale, there are so many great novels about women who rebelled against a male-dominated society that sought to control them. Many of these women were inspired by real people, and in turn they inspired many more women to break the mould and challenge social norms.

But for every Elizabeth Bennet, there's an empowering female character who, for whatever reason, hasn't received the same recognition. Let's take another look back at those pioneering women, shall we? 

Virginia Woolf's 1928 novel is a marvelously strange feminist classic, the story of aristocratic 16th-century poet Orlando, who changes sex overnight from male to female. As simply "Orlando," she lives through several centuries of English literary history as a woman, while frequently cross-dressing as a man. 

Woolf's inspiration for the character of Lady Orlando was Vita Sackville-West, her friend and sometimes-lover who was well-known as being bisexual, gender fluid, and polyamorous at a time when all of those traits were major social taboos. The novel is a celebration of androgyny, a critique of misogyny and gender roles, and an empowering feminist treatment of British literary history. 

This book should be read far more widely and praised far more often, period.

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We've all seen the Disney movie adaptation featuring Eddie Murphy as a talking dragon, but the story of Mulan is actually really, really old. The Ballad of Mulan is a legend dating back to the 6th century, about a young woman who dresses as a man and joins the Chinese army because her father is too old and weak to fight. 

She spends 12 years fighting in the army, developing a high reputation for her heroism. She is skilled in martial arts and swordfighting, and when she retires she is offered a government post. She declines and returns to her hometown, but first reveals to her fellow soldiers that she is a woman--and they accept her completely. Gotta love a happy ending.

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Although Ray Bradbury's classic 1953 science fiction novel is focused on the "firefighter" Guy Montag, whose job is to burn books in a dystopian version of America, his neighbor Clarisse McClellan is the reason Montag even begins to question his society. 

Clarisse, who was played by Julie Christie in the 1966 film adaptation, is a teenager but wise beyond her years. She is shunned by her peers because of her disinterest in mindless entertainment and fascination with instead exploring nature and ideas. Her insatiable curiosity sparks Montag's own curiosity, and her tragic death in a car accident near the beginning of the novel is a truly heartbreaking twist.

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In the stage musical and film adaptation of Les Misrables, ponine Thnardier is a rather simple girl, a young lover with a one-way crush on the handsome young revolutionary Marius.

In Victor Hugo's 1862 novel, however, ponine is a scrappy, street-smart lady who sometimes plays dirty in order to get what she wants. She's not always heroic in her deeds--she bullies Cosette and helps her father in his racketeering--but she becomes more honest as the novel progresses. 

More importantly, she is a three-dimension female character, not romanticized or objectified but full of the internal conflicts that form a human being, and sadly that's a rare and valuable thing in classic literature.

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Gabriel Garca Mrquez's 1967 novel is considered a landmark in the history of Latin American literature, but its female characters don't get enough credit. rsula Iguarn is the matriarch of the Buenda family and can be seen as the unsung all-star of the novel, since she keeps rebuilding the Buenda house again and again over the course of her 150-year lifespan. 

The lady keeps on trucking while the rest of her family keeps dropping like flies. rsula for the win.

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Toni Morrison's 1973 novel contrasts the lives of two black women in post-World War One Ohio, one named Nel and the other Sula. While Nel ends up choosing a conventional life for a woman, marrying and having children in their town, Sula chooses to completely abandon all social conventions, and spends a decade traveling America having various affairs and adventures. 

Sula is also an androgynous character, as represented by her birthmark shaped like a "stemmed rose"--which has both phallic and vaginal, masculine and feminine resonance. 

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The 1985 science fiction novel by Orson Scott Card tells the story of a boy genius named Ender Wiggin, who lives on a futuristic Planet Earth where children must be trained in combat games to fight invading aliens.

The character of Ender gets much of the credit for his incredible abilities, both in the novel and in sci-fi fandom, but his sister Valentine is equally awesome. She uses her profound empathy to inspire Ender to spring into action, as well as to calm down her other brother Peter, a brilliant psychopath who aims to take over Earth's government. She ends up becoming a massively powerful politician herself. Pretty good for a kid who would have been in elementary school in our day.

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Boldly sexual and well-known for her promiscuity, Brett, Lady Ashley is one of the most memorable characters in all of Hemingway's work. The novel about young Americans and Brits living in Europe was based on Hemingway's own experiences, and specifically the character of Brett was inspired by his infatuation with the Lady Duff, another member of his "Lost Generation" circle. 

Hemingway's portrayal has misogynistic undertones--he implies is it her fault that all the men around her seem to behave so self-destructively in her presence--yet the Lady Ashley remains confident, independent, and adventurous in a world that was certainly not built for her.

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Elizabeth Gaskell's 1853 novel focused on what sounds like a stuffy subject: the relationship between mill owners and laborers in Northern England during industrialization. But it does so from an interesting perspective: that of a young woman, Margaret Hale, who is sympathetic to the exploited workers and willing to fight for social change.

At a time when women were expected to be seen and not heard, Margaret Hale creates a lot of noise, clashing with wealthy mill owner John Thornton and pressuring him to listen to the demands of his striking workers. Thornton falls in love with Hale and they eventually reconcile, but only after Thornton has changed his attitude and gained a great deal of empathy, simply proving Margaret's integrity and its power to change others.

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Charlotte Lucas isn't the most beautiful, or the most lively character in Jane Austen's 1813 novel Pride and Prejudice. But that's part of what makes her worthy of admiration. 

She doesn't have the social advantages that Elizabeth has that allow her to court the handsome and wealthy Mr. Darcy. Instead, she's a realist to the core, and seeing that at age 27, the most ideal way to move forward with her life is to marry Mr. Collins, she goes ahead and does it. Girl got married. Deal with it. 

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Everyone talked about Liv Tyler's turn as Arwen in the Peter Jackson film adaptations, but the shieldmaiden of Rohan is arguably an even bigger hero. owyn's character is pretty much summed up with this quote from Return of the King:

"Shall I always be left behind when the Riders depart, to mind the house while they win renown, and find food and beds when they return?"

The answer, as we know from the novel and the film adaptation, is no friggin' way. owyn fights valiantly at the climactic battle of Return of the King, eventually coming face to face with the Witch-King, Lord of Ring Wraiths. While he laughs at her, she stabs him in the face, killing him and thus fulfilling a thousand-year-old prophecy that "not by the hand of man" would the Witch-King be slain. Yeah she did.

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Did you know that Charlotte Bront's Jane Eyre had a prequel? Well one does exist, although it wasn't written by Bront, and was published over 100 years after the original. 

Wide Sargasso Sea (1966) was the work of Dominican-born British writer Jean Rhys, who transformed a minor character from the original work, Mr. Rochester's first wife Bertha Mason, into the protagonist of the prequel. In Jane Eyre, Mason is described as "the madwoman in the attic"--a woman that Mr. Rochester never loved and cannot be with because she is mentally unstable.

Rhys' intention was to reveal Bertha Mason as actually having been driven to mental instability by the oppressiveness of her husband, and of the society that forced her to marry him. It's a much more nuanced depiction, and one that challenges the way male-dominated culture has tried to silence independent or strong-willed women by categorizing them as mentally ill.  

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British actress Emma Thompson played the elder Schlegel sister in the 1993 film adaptation of E.M. Forster's novel, and naturally we picture her when we think of the character.

Kind and thoughtful, compassionate but careful, Margaret is the perfect counterweight to her younger sister Helen's unbridled romantic energy. At the same time, Margaret is courted by a much more self-centered person, the wealthy Henry Wilcox, and agrees to marry him because she is so able to see the good in him. 

When Margaret becomes torn between loyalty to her sister Helen, and to Henry--who has the ability to save the life of Helen's lover--she does something remarkable and chooses both. She not only saves her marriage, she also empowers her husband to gain the empathy he needs to see the value in helping others. Now that's how it's done.

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Adriana is one of the lesser-known female characters in Shakespeare's plays, but she's one of my personal favorites and the unsung hero of this early farce. She believes that her husband Antipholus left her abruptly (he actually was mistakenly arrested and put in jail), and in response goes on a rollercoaster of emotions as she attempts to understand why. 

She's a deeply faithful and loving partner, and her feelings of doubt and insecurity are all too relatable. She also has what I think is one of the most romantic monologues in all of Shakespeare, where she describes how she feels about her husband:

For know, my love, as easy mayest thou fall / A drop of water in the breaking gulf, / And take unmingled thence that drop again / Without addition or diminishing, / As take from me thyself and not me too (2.2.120-124).

In other words, trying to forget her husband would be as impossible as putting a drop of water in the ocean and then attempting to take it out again; that's how interconnected they are after 20 years of marriage.   

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The 5-year-old hero of Roald Dahl's 1988 classic is no average kid. Despite being surrounded by monstrous adults who attempt to bully her into conformity, she stays strong and stays weird. 

Not only does she have telekinetic powers that allow her to gain the upper hand and become a folk hero to her classmates, she's also wicked smart with a sense of humanity way beyond her years. Let's go watch the movie version, shall we?

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Let's be honest, the Hardy Boys had zero chill. Besides, why should boys have all the detective-mystery fun? Nancy Drew first appeared in 1930, meaning she has been inspiring girls to go sleuthing for close to a century now, a true collective effort with countless authors contributing to the series over the decades.

Her fanbase includes such accomplished women as Supreme Court Justice Sandra Day O'Connor, Oprah Winfrey, and Hilary Rodham Clinton.  

Favorite quote: Ive fought imaginary elves that were stronger than you! 

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The Famous Five was a series of popular children's novels published in the 1940s by British author Enid Blyton. The stories were about a group of children going on adventures in their seaside town during the summer holidays, and the tales were fairly edgy for their time, sometimes involving encounters with criminals.

The best part? Their unofficial leader was George, a tomboy whose given name was Georgina but insisted on being treated exactly the same as the boys treated each other. Headstrong and courageous, she took absolutely no crap from anyone, especially the boys.

"I shan't answer to the name Georgina!" she would say when anyone got it twisted. Blyton eventually revealed that the character of George was based on herself.

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The Color Purple is Alice Walker's Pulitzer-Prize-winning novel about African-American women living in rural Georgia during the Great Depression, under incredibly oppressive circumstances. 

Sofia, played by Oprah Winfrey in the film adaptation, is an imposing person who, despite her social position as a poor black woman, refuses to let her husband beat or abuse her, and in one instance fights back. She is put in jail for this act of self-defense, but maintains her innocence. Oprah definitely did the character justice in the film, receiving an Oscar nomination for her performance.

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Moll Flanders goes a long way over the course of Daniel Defoe's 1722 novel. Born to a mother who is on death row, and forced to become a servant as a teenager, she never has much of a chance to take the easy way through life. Instead, in order to become an independent person, she gains a fortune by posing as a rich heiress and thus conning men who are actually rich into marrying her. 

Moll Flanders is by no means perfect, but she uses her wits to get what she wants and restart her life a dozen times over in a world where women had almost no say in how they wanted to live their lives. Ya do what ya gotta do.

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Nathaniel Hawthorne's 1850 novel "The Scarlet Letter" is the fictional story of Hester Prynne, a young woman in Puritan Boston in 1642 who gives birth to a child whose father is unknown. As a punishment, she is publicly humiliated on a platform in the middle of Boston and forced to wear a letter "A" for "Adulteress" on her blouse for the rest of her life. 

Prynne is repeatedly humiliated and insulted by her fellow townsfolk throughout the novel, but she never buckles under the pressure and keeps her mystery partner a secret. She raises a child as a single mom, and the supposed "adultery" that she committed happened with the town's priest... after her husband had abandoned her to become a sailor. No matter what the Puritans say, we know that Hester Prynne is just a young lover who was mistreated by her community.

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