Tag Archives: probability

The Birthday Problem


I first came across the Birthday Problem in my first few days of my first year of secondary school in London [that would be 6th grade for my US readers]. My mathematics teacher at the time realized the need to discuss abstract problems in concrete terms, especially statistics and probability. So, he wowed many of us — in a class of close to 30 kids — by firmly stating that there was a better than even chance that two of us shared the same birthday. In a class of 30, the actual probability is 60 percent, and rises to close to 100 percent is a group of only 60.

Startlingly, two in our class did indeed share the same birthday. How could that be possible, I wondered?

Well, the answer is grounded in the simple probability of large populations. But, it is also colored by our selective biases to remember “remarkable” coincidences and to ignore the much, much larger number of instances where there is no coincidence at all.

From the Washington Post.

Mathematician Joseph Mazur was in the back of a van snaking through the mountains of Sardinia when he heard one of his favorite coincidence stories. The driver, an Italian language teacher named Francesco, told of meeting a woman named Manuela who had come to study at his school. Francesco and Manuela met for the first time in a hotel lobby, and then went to have coffee.

They spoke for an hour, getting acquainted, before the uncomfortable truth came out. Noting Manuela’s nearly perfect Italian, Francesco finally asked why she decided to come to his school.

“She said, ‘Italian? What are you talk about? I’m not here to learn Italian,’” Mazur relates. “And then it dawned on both of them that she was the wrong Manuela and he was the wrong Francesco.” They returned to the hotel lobby where they had met to find a different Francesco offering a different Manuela a job she didn’t want or expect.

The tale is one of the many stories that populate Mazur’s new book, “Fluke,” in which he explores the probability of coincidences.

Read the entire article here.

Image: The computed probability of at least two people sharing a birthday versus the number of people. Courtesy: Rajkiran g / Wikipedia. CC BY-SA 3.0.

Fate Isn’t All That It’s Cracked Up to Be

If you believe in the luck of the draw, the turn of a card, the spin of a wheel; if you believe in the leaves in your teacup, the lines on your palm, or the numbers in your fortune cookie; if you believe in fate or a psychic or the neighbor’s black cat, then you are all the poorer for it — perhaps not spiritually, but certainly financially.

From the Telegraph:

Strange as it sounds, a serious study has been undertaken by academics into the link between people’s propensity to trust in luck, or fate – and their financial success.

And it has concluded the less faith someone places in luck, fate or some other “external factor”, the more wealth they are likely to accumulate.

Some might say the conclusion is commonsense but the report – produced by three academics at the University of Mebourne in Australia – even came up with a figure of AUS$150,000 (£82,000), which was the difference over four years between “households who believe fate will determine their future” and “households that believe they can shape their own destiny.”

The report, here, titled “Locus of control and savings”, splits psychological profiles into two groups, those with either an “internal” or “external” “locus” of control. The latter are people who believe that fate, or luck – or other people – are the determining force in shaping their lives. Those with an “internal locus of control” are those who are “strong believers in their ability to shape their own destiny.”

The survey then linked pshychological measures of behaviour to national savings data. “We find that households in which the reference person has an internal locus of control save more both in terms of levels and as a percentage of their permanent incomes than do households with external reference persons.”

It arrived at a precise financial measure, saying: “over a four year period households with a strong sense of shaping one’s destiny are on average $150,000 better off, and save 7.7% more of their income.”

The authors claimed that although their work relied on Australian data, it would reflect trends in other developed economies.

The work is one of a growing number of studies into what motivates saving, the type of people most likely to save – and how governments can stimulate more saving.

Read the entire article here.


One of our favorite thinkers (and authors) here at theDiagonal is Nassim Taleb. His new work entitled Antifragile expands on ideas that he first described in his bestseller Black Swan.

Based on humanity’s need to find order and patterns out of chaos, and proclivity to seek causality where none exists we’ll need several more books from him before his profound and yet common-sense ideas sink in. In his latest work, Taleb shows how the improbable and unpredictable lie at the foundation of our universe.

[div class=attrib]From the Guardian:[end-div]

Now much does Nassim Taleb dislike journalists? Let me count the ways. “An erudite is someone who displays less than he knows; a journalist or consultant the opposite.” “This business of journalism is about pure entertainment, not the search for the truth.” “Most so-called writers keep writing and writing with the hope, some day, to find something to say.” He disliked them before, but after he predicted the financial crash in his 2007 book, The Black Swan, a book that became a global bestseller, his antipathy reached new heights. He has dozens and dozens of quotes on the subject, and if that’s too obtuse for us non-erudites, his online home page puts it even plainer: “I beg journalists and members of the media to leave me alone.”

He’s not wildly keen on appointments either. In his new book, Antifragile, he writes that he never makes them because a date in the calendar “makes me feel like a prisoner”.

So imagine, if you will, how keenly he must be looking forward to the prospect of a pre-arranged appointment to meet me, a journalist. I approach our lunch meeting, at the Polytechnic Institute of New York University where he’s the “distinguished professor of risk engineering”, as one might approach a sleeping bear: gingerly. And with a certain degree of fear. And yet there he is, striding into the faculty lobby in a jacket and Steve Jobs turtleneck (“I want you to write down that I started wearing them before he did. I want that to be known.”), smiling and effusive.

First, though, he has to have his photo taken. He claims it’s the first time he’s allowed it in three years, and has allotted just 10 minutes for it, though in the end it’s more like five. “The last guy I had was a fucking dick. He wanted to be artsy fartsy,” he tells the photographer, Mike McGregor. “You’re OK.”

Being artsy fartsy, I will learn, is even lower down the scale of Nassim Taleb pet hates than journalists. But then, being contradictory about what one hates and despises and loves and admires is actually another key Nassim Taleb trait.

In print, the hating and despising is there for all to see: he’s forever having spats and fights. When he’s not slagging off the Nobel prize for economics (a “fraud”), bankers (“I have a physical allergy to them”) and the academic establishment (he has it in for something he calls the “Soviet-Harvard illusion”), he’s trading blows with Steven Pinker (“clueless”), and a random reviewer on Amazon, who he took to his Twitter stream to berate. And this is just in the last week.

And yet here he is, chatting away, surprisingly friendly and approachable. When I say as much as we walk to the restaurant, he asks, “What do you mean?”

“In your book, you’re quite…” and I struggle to find the right word, “grumpy”.

He shrugs. “When you write, you don’t have the social constraints of having people in front of you, so you talk about abstract matters.”

Social constraints, it turns out, have their uses. And he’s an excellent host. We go to his regular restaurant, a no-nonsense, Italian-run, canteen-like place, a few yards from his faculty in central Brooklyn, and he insists that I order a glass of wine.

“And what’ll you have?” asks the waitress.

“I’ll take a coffee,” he says.

“What?” I say. “No way! You can’t trick me into ordering a glass of wine and then have coffee.” It’s like flunking lesson #101 at interviewing school, though in the end he relents and has not one but two glasses and a plate of “pasta without pasta” (though strictly speaking you could call it “mixed vegetables and chicken”), and attacks the bread basket “because it doesn’t have any calories here in Brooklyn”.

But then, having read his latest book, I actually know an awful lot about his diet. How he doesn’t eat sugar, any fruits which “don’t have a Greek or Hebrew name” or any liquid which is less than 1,000 years old. Just as I know that he doesn’t like air-conditioning, soccer moms, sunscreen and copy editors. That he believes the “non-natural” has to prove its harmlessness. That America tranquillises its children with drugs and pathologises sadness. That he values honour above all things, banging on about it so much that at times he comes across as a medieval knight who’s got lost somewhere in the space-time continuum. And that several times a week he goes and lifts weights in a basement gym with a bunch of doormen.

He says that after the financial crisis he received “all manner of threats” and at one time was advised to “stock up on bodyguards”. Instead, “I found it more appealing to look like one”. Now, he writes, when he’s harassed by limo drivers in the arrival hall at JFK, “I calmly tell them to fuck off.”

Taleb started out as a trader, worked as a quantitative analyst and ran his own investment firm, but the more he studied statistics, the more he became convinced that the entire financial system was a keg of dynamite that was ready to blow. In The Black Swan he argued that modernity is too complex to understand, and “Black Swan” events – hitherto unknown and unpredicted shocks – will always occur.

What’s more, because of the complexity of the system, if one bank went down, they all would. The book sold 3m copies. And months later, of course, this was more or less exactly what happened. Overnight, he went from lone-voice-in-the-wilderness, spouting off-the-wall theories, to the great seer of the modern age.

Antifragile, the follow-up, is his most important work so far, he says. It takes the central idea of The Black Swan and expands it to encompass almost every other aspect of life, from the 19th century rise of the nation state to what to eat for breakfast (fresh air, as a general rule).

[div class-attrib]Read the entire article following the jump.[end-div]

[div class=attrib]Image: Black Swan, the movie, not the book by the same name by Nassim Taleb. Courtesy of Wkipedia.[end-div]

How Will You Die?

Bad news and good news. First, the bad news. If you’re between 45-54 years of age your cause of death will most likely be heart disease, that is, if you’re a male. If you are a female on the other hand, you’re more likely to fall prey to cancer. And, interestingly you are about 5 times more likely to die falling down stairs than from (accidental) electrocution. Now the good news. While the data may give us a probabilistic notion of how we may perish, no one (yet) knows when.

More vital statistics courtesy of this macabre infographic derived from data of National Center for Health Statistics and the National Safety Council.

Chance as a Subjective or Objective Measure

[div class=attrib]From Rationally Speaking:[end-div]

Stop me if you’ve heard this before: suppose I flip a coin, right now. I am not giving you any other information. What odds (or probability, if you prefer) do you assign that it will come up heads?

If you would happily say “Even” or “1 to 1” or “Fifty-fifty” or “probability 50%” — and you’re clear on WHY you would say this — then this post is not aimed at you, although it may pleasantly confirm your preexisting opinions as a Bayesian on probability. Bayesians, broadly, consider probability to be a measure of their state of knowledge about some proposition, so that different people with different knowledge may correctly quote different probabilities for the same proposition.

If you would say something along the lines of “The question is meaningless; probability only has meaning as the many-trials limit of frequency in a random experiment,” or perhaps “50%, but only given that a fair coin and fair flipping procedure is being used,” this post is aimed at you. I intend to try to talk you out of your Frequentist view; the view that probability exists out there and is an objective property of certain physical systems, which we humans, merely fallibly, measure.

My broader aim is therefore to argue that “chance” is always and everywhere subjective — a result of the limitations of minds — rather than objective in the sense of actually existing in the outside world.

[div class=attrib]Much more of this article here.[end-div]

[div class=attrib]Image courtesy of Wikipedia.[end-div]

Book Review: The Drunkard’s Walk: How Randomness Rules Our Lives. Leonard Mlodinow

Leonard Mlodinow weaves a compelling path through the world of statistical probability showing us how the laws of chance affect our lives on personal and grande scales. Mlodinow skillfully illustrates randomness and its profound implications by presenting complex mathematical constructs in language for the rest of us (non-mathematicians), without dumbing-down this important subject.

The book defines many of the important mathematical concepts behind randomness and exposes the key fallacies that often blind us as we wander through life on our “drunkard’s walk”. The law of large numbers, the prosecutor’s fallacy, conditional probability, the availability bias and bell curves were never so approachable.

Whether it’s a deluded gambler, baseball star on a “winning streak” or a fortunate CEO wallowing in the good times, Mlodinow debunks the common conceptions that skill, planning and foresight result in any significant results beyond pure chance. With the skill of a storyteller Mlodinow shows us how polls, grades, ratings and even measures of corporate success are far less objective and reliable than we ought to believe. Lords of Wall Street take notice, the secrets of your successes are not all that they seem.