## Posts Tagged ‘**Probability**’

## “Mathematics, rightly viewed, possesses not only truth, but supreme beauty”*…

Maryam Mirzakhani did not enjoy mathematics to begin with. She dreamed of being an author or politician, but as a top student at her all-girls school in Tehran she was still disappointed when her first-year maths exam went poorly. Her teacher believed her – wrongly – to have no particular affinity with the subject.

Soon that would all change. “My first memory of mathematics is probably the time [my brother] told me about the problem of adding numbers from 1 to 100,” she recalled later. This was the story of Carl Gauss, the 18th-century genius whose schoolteacher set him this problem as a timewasting exercise – only for his precocious pupil to calculate the answer in a matter of seconds.

The obvious solution is simple but slow: 1+2+3+4. Gauss’s solution is quicker to execute, and far more cunning. It goes like this: divide the numbers into two groups: from 1 to 50, and from 51 to 100. Then, add them together in pairs, starting with the lowest (1) and the highest (100), and working inwards (2+99, 3+98, and so on). There are 50 pairs; the sum of each pair is 101; the answer is 5050. “That was the first time I enjoyed a beautiful solution,” Mirzakhani told the Clay Mathematics Institute in 2008.

Since then, her appreciation for beautiful solutions has taken her a long way from Farzanegan middle school. At 17 she won her first gold medal at the International Mathematics Olympiad. At 27 she earned a doctorate from Harvard University. The Blumenthal Award and Satter Prize followed, and in 2014 she became the first woman to be awarded the Fields Medal, the highest honour a mathematician can obtain.

Before this particular brand of wonder became perceptible to Mirzakhani, she experienced feelings many of us can relate to: to the indifferent, her subject can seem “cold”, even “pointless”. Yet those who persist will be rewarded with glimpses of conceptual glory, as if gifted upon them by a capricious god: “The beauty of mathematics,” she warned, “only shows itself to more patient followers.”

This concept of “beauty” found in maths has been referred to over centuries by many others; though, like beauty itself, it is notoriously difficult to define…

For an experienced mathematician, the greatest equations are beautiful as well as useful. Can the rest of us see what they see? “What makes maths beautiful?”

[From *The New Humanist*, via the ever-illuminating 3 Quarks Daily]

Maryam Mirzakhani died last Friday, a victim of breast cancer; she was 40. As Peter Sarnak (a mathematician at Princeton University and the Institute for Advanced Study) said, her passing is “a big loss and shock to the mathematical community worldwide.” See also here.

* Bertrand Russell, *A History of Western Philosophy*

###

**As we accede to awe,** we might spare a thought for Andrey (Andrei) Andreyevich Markov; he died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of *Monopoly* can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological and social sciences.

## Fun with numbers!…

Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”

The event was the Gathering for Gardner [see

here], a convention held every two years in Atlanta, Georgia, uniting mathematicians, magicians and puzzle enthusiasts. The audience was silent as they pondered the question.“The first thing you think is ‘What has Tuesday got to do with it?'” said Foshee, deadpan. “Well, it has everything to do with it.” And then he stepped down from the stage.

Read the full story of the conclave– held in honor of the remarkable Martin Gardner, who passed away last year, and in the spirit of his legendary “Mathematical Games” column in *Scientific American*— in * New Scientist*… and find the answer to Gary’s puzzle there– or after the smiling professor below.

“I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”… readers may hear a Bayesian echo of **the Monty Hall Problem** on which (R)D has mused before:

The first thing to remember about probability questions is that everyone finds them mind-bending, even mathematicians. The next step is to try to answer a similar but simpler question so that we can isolate what the question is really asking.

So, consider this preliminary question: “I have two children. One of them is a boy. What is the probability I have two boys?”

This is a much easier question: The way Foshee meant it is, of all the families with one boy and exactly one other child, what proportion of those families have two boys?

To answer the question you need to first look at all the equally likely combinations of two children it is possible to have: BG, GB, BB or GG. The question states that one child is a boy. So we can eliminate the GG, leaving us with just three options: BG, GB and BB. One out of these three scenarios is BB, so the probability of the two boys is 1/3.

Now we can repeat this technique for the original question. Let’s list the equally likely possibilities of children, together with the days of the week they are born in. Let’s call a boy born on a Tuesday a BTu. Our possible situations are:

* When the first child is a BTu and the second is a girl born on any day of the week: there are seven different possibilities.

* When the first child is a girl born on any day of the week and the second is a BTu: again, there are seven different possibilities.

* When the first child is a BTu and the second is a boy born on any day of the week: again there are seven different possibilities.

* Finally, there is the situation in which the first child is a boy born on any day of the week and the second child is a BTu – and this is where it gets interesting.There are seven different possibilities here too, but one of them – when both boys are born on a Tuesday – has already been counted when we considered the first to be a BTu and the second on any day of the week. So, since we are counting equally likely possibilities, we can only find an extra six possibilities here.

Summing up the totals, there are 7 + 7 + 7 + 6 = 27 different equally likely combinations of children with specified gender and birth day, and 13 of these combinations are two boys. So the answer is 13/27, which is very different from 1/3.

It seems remarkable that the probability of having two boys changes from 1/3 to 13/27 when the birth day of one boy is stated – yet it does, and it’s quite a generous difference at that. In fact, if you repeat the question but specify a trait rarer than 1/7 (the chance of being born on a Tuesday), the closer the probability will approach 1/2.

[See **UPDATE**, below]

**As we remember, with Laplace, that “the theory of probabilities is at bottom nothing but common sense reduced to calculus,”** we might ask ourselves what the odds are that on this date in 1964 the World’s Largest Cheese would be manufactured for display in the Wisconsin Pavilion at the 1964-65 World’s Fair. The 14 1/2′ x 6 1/2′ x 5 1/2′, 17-ton cheddar original– the product of 170,000 quarts of milk from 16,000 cows– was cut and eaten in 1965; but a replica was created and put on display near Neillsville, Wisconsin… next to **Chatty Belle**, the World’s Largest Talking Cow.

The replica on display (*source*)

**UPDATE:** reader Jeff Jordan writes with a critique of the reasoning used above to solve Gary Foshee’s puzzle:

For some reason, mathematicians and non-mathematicians alike develop blind

spots about probability problems when they think they already know the

answer, and are trying to convince others of its correctness. While I agree

with most of your analysis, it has one such blind spot. I’m going move

through a progression of variations on another famous conundrum, trying to

isolate these blind spots and eventually get the point you overlooked.Bertrand’s Box Paradox: Three identical boxes each have two coins inside:

one has two gold coins, one has two silver coins, and one has a silver coin

and a gold coin. You open one and pull out a coin at random, without seeing

the other. It is gold. What is the probability the other coin is the same

kind?A first approach is to say there were three possible boxes you could pick,

but the information you have rules one out. That leaves two that are still

possible. Since you were equally likely to pick either one before picking a

coin, the probability that this box is GG is 1/2. A second approach is that

there were six coins that were equally likely, and three were gold. But two

of them would have come out of the GG box. Since all three were equally

likely, the probability that this box is GG is 2/3.This appears to be a true paradox because the “same” theoretical approach –

counting equally likely cases – gives different answers. The resolution of

that paradox – and the first blind spot – is that this is an incorrect

theoretical approach to solving the problem. You never want to merely count

cases, you want to sum the probabilities that each case would produce the

observed result. Counting only works when each case that remains possible

has the same chance of producing the observed result. That is true when you

count the coins, but not when you count the boxes. The probability of

producing a gold coin from the GG box is 1, from the SS box is 0, and from

the GS box is 1/2. The correct answer is 1/(1+0+1/2)=2/3. (A second blind

spot is that you don’t “throw out” the impossible cases, you assign them a

probability of zero. That may seem like a trivial distinction, but it helps

to understand what probabilities other than 1 or 0 mean.)This problem is mathematically equivalent to the original Monty Hall

Problem: You pick Door #1 hoping for the prize, but before opening it the

host opens Door #3 to show that it is empty. Given the chance, what is the

probability you win by switching to door #2? Let D1, D2, and D3 represent

where the prize is. Assuming the host won’t open your door, and knows where

the prize is so he always opens an empty door, then the probability D2 would

produce the observed result is 1, that D3 would is 0, and that D1 is …

well, let’s say it is 1/2. Just like before, the probability D2 now has the

prize is 1/(1+0+1/2)=2/3.Why did I waffle about the value of P(D1)? There was a physical difference

with the boxes that produced the explicit result P(GS)=1/2. But here the

difference is logical (based on the location of the prize) and implicit. Do

we really know the host would choose randomly? In fact, if the host always

opens Door #3 if he can, then P(D1)=1 and the answer is 1/(1+0+1)=1/2. Or if

he always opens Door #2 if he can, P(D1)=0 and the answer is 1/(1+0+0)=1.

But if we observe that the host opened Door #2 and assume those same biases,

the results reverse.To answer the question, we must assume a value for P(D1). Assuming anything

other than P(D1)=1/2 implies a bias on the part of the host, and a different

answer if he opens Door #2. So all we can assume is P(D1)=1/2, and the

answer is again 2/3. That is also the answer if we average the results over

many games with the same host (and a consistent bias, whatever it is). The

answer most “experts” give is really that average, and it is a blind spot

that they are not using all the information they have in the individual

case.We can make the Box Paradox equivalent to this one by making the random

selection implicit. Someone looks in the chosen box, and picks out a gold

coin. The probability is 2/3 that there is another gold coin if that person

picks randomly, 1/2 if that person always prefers a gold coin, and 1 if that

person always prefers a silver one. Without knowing the preference, we can

only assume this person is unbiased and answer 2/3. Over many experiments,

it will also average out to 2/3 regardless of the bias. And this person

doesn’t even have to show the coin. If we assume he is truthful (and we can

only assume that), the answers are the same if he just says “One coin is

gold.”Finally, make a few minor changes to the Box Paradox. Change “silver” to

“bronze.” Let the coins be minted in different years, so that the year

embossed on them is never the same for any two. Add a fourth box so that one

box has an older bronze coin with a younger gold coin, and one has a younger

bronze coin with an older gold coin. Now we can call the boxes BB, BG, GB,

and GG based on this ordering. When our someone says “One coin is bronze,”

we can only assume he is unbiased in picking what kind of coin to name, and

the best answer is 1/(1+1/2+1/2+0)=1/2. If there is a bias, it could be

1/(1+1+1+0)=1/3 or 1/(1+0+0+0)=1, but we can’t assume that. Gee, this sounds

oddly familiar, except for the answer. :)The answer to all of Gary Foshee’s questions is 1/2. His blind spot is that

he doesn’t define events, he counts cases. An event a set of outcomes, not

an outcome itself. The sample space is the set of all possible outcomes. An

event X must be defined by some property such that every outcome in X has

that property, *and* every outcome with the property is in X. The event he

should use as a condition is not “this family includes a boy (born on a

Tuesday)”, it is “The father of this family chooses to tell you one of up to

two facts in the form ‘my family includes a [gender] (born on a [day]).'”

Since most fathers of two will have two different facts of that form to

choose from, Gary Foshee should have assigned a probability to each, not

merely counted the families that fit the description. The answer is then

(1+12P)/(1+26P), where P is the probability he would tell us “one is a boy

born on a Tuesday” when only one of his two children fit that description.

The only value we can assume for P is 1/2, making the answer

(1+6)/(1+13)=1/2. Not P=1 and (1+12)/(1+26)=13/27.And the blind spot that almost all experts share, is that this means the

answer to most expressions of the simpler Two Child Problem is also 1/2. It

can be different, but only if the problem statement makes two or three

points explicit:1) Whatever process led to your knowledge of one child’s gender had access

to both children’s genders (and days of birth).

2) That process was predisposed to mention boys over girls (and Tuesdays

over any other day).

3) That process would never mention facts about both children.When Gary Foshee tells you about one of his kids, #2 is not satisfied. He

probably had a choice of two facts to tell you, and we can’t assume he was

biased towards “boy born on Tuesday.” Just like Monty Hall’s being able to

choose two doors changes the answer from 1/2 to 2/3, Gary Foshee’s being

able to choose two facts changes the answer from 13/27 to 1/2. It is only

13/27 if he was forced to mention that fact, which is why that answer is

unintuitive.

Other readers are invited to contribute their thoughts.

## By the numbers…

Mark Twain quotes Disraeli: “There are three kinds of lies: lies, damned lies, and statistics”; H.G. Wells avers that “Satan delights equally in statistics and in quoting scripture”; but the remarkable **Hans Rosling** begs to differ…

Rosling, a physician and medical researcher who co-founded Médecins sans Frontièrs (Doctors without Borders) Sweden and the **Gapminder Foundation** (with his son and daughter-in-law), and developed the **Trendalyzer** software that represents national and global statistics as animated interactive graphics (e.g., **here**), ha become **a superstar** on the lecture circuit. He brings his unique insight and approach to the BBC with *The Joy of Stats*…

~~It’s above at full length, so takes a while to watch in toto~~— but odds are that one will enjoy it! [UPDATE: since this post was published, the full version has been rendered “private”; unless and until it’s reposted in full, the taste above will have to do. Readers in the UK (or readers with VPNs that terminate in the UK) can see the full show soon after it airs on BBC Four on Thursday the 13th on the **BBC iPlayer**. As a further consolation, **here** is statistician Andrew Gelman’s “Five Books” interview– his choice of the five best books on statistics– for *The Browser*. ]

**As we realize that sometimes we can, after all, count on it,** we might recall that it was on this date in 1776 that Thomas Paine (originally anonymously) published his case for the independence of the American Colonies, “Common Sense”… and after all, as Pierre-Simon, marquis de Laplace pointed out (in 1820), “the theory of probabilities is at bottom nothing but common sense reduced to calculus.”

*source: University of Indiana*

## What you do know can hurt you…

From the ever-entertaining **xkcd**, a behavioral analog to **the Monty Hall Problem** (and **the variation** considered here a couple of weeks ago)…

**As we reconsider the odds,** we might recall that it was on this date in 1777 that Swiss mathematician, physicist, and astronomer Johann Heinrich Lambert died in Berlin. Lambert, who was only 49 when he passed, made a number of contributions to scientific knowledge; but he is probably best remembered for the first proof (in 1768) that pi is irrational (that’s to say, can’t be expressed as the quotient of two integers).

## Odds are that your bank balance is…

From * Technology Review*:

A computer chip that performs calculations using probabilities, instead of binary logic, could accelerate everything from online banking systems to the flash memory in smart phones and other gadgets.

Rewriting some fundamental features of computer chips, Lyric Semiconductor has unveiled its first “probability processor,” a silicon chip that computes with electrical signals that represent chances, not digital 1s and 0s.

“We’ve essentially started from scratch,” says Ben Vigoda, CEO and founder of the Boston-based startup. Vigoda’s PhD thesis underpins the company’s technology. Starting from scratch makes it possible to implement statistical calculations in a simpler, more power efficient way, he says…

Read the full story **here**.

**As we remind ourselves that dealing with our banks was already a crap-shoot,** we might recall that it was on this date in 79 CE, the feast day of Vulcan, the Roman god of fire, that Mount Vesuvius began to stir– in preparation for

**the eruption that, two days later, destroyed the cities of Pompeii and Herculaneum**.

Fresco of Bacchus and Agathodaemon with Mount Vesuvius, as seen in Pompeii’s House of the Centenary (*source*)

## Birdbrains…

Readers will recall “The Monty Hall Problem”:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

As explained in “**Riddle Me This…**,” the Bayesian peculiarities of the answer are not always intuitively obvious. Indeed, as ** Discover reports**,

Over the years, the problem has ensnared countless people, including professional mathematicians. But not, it seems, pigeons. Walter Hebranson and Julia Schroder showed that, after some training, the humble pigeon can learn the best tactic for the Monty Hall Problem, switching from their initial choice almost every time. Amazingly, humans who get similar extensive practice never develop the optimal strategies that the pigeons pick up.

The original paper “Are birds smarter than mathematicians? Pigeons (Columba livia) perform optimally on a version of the Monty Hall Dilemma” from the *Journal of Comparative Psychology* is **here**… but **the Discover article** is a good– and for the humans among us, chastening– summary.

**As we retreat to our fingers for counting,** we might recall that it was on this date in 1833 that the first tax-supported public library was founded, in Peterborough, NH. The original collection consisted of about 100 books and was kept in Smith & Thompson’s General Store, which also housed the Post Office.

## Playing the odds…

*A P value is the probability of an observed (or more extreme) result arising only from chance.*

It’s science’s dirtiest secret: The “scientific method” of testing hypotheses by statistical analysis stands on a flimsy foundation. Statistical tests are supposed to guide scientists in judging whether an experimental result reflects some real effect or is merely a random fluke, but the standard methods mix mutually inconsistent philosophies and offer no meaningful basis for making such decisions. Even when performed correctly, statistical tests are widely misunderstood and frequently misinterpreted. As a result, countless conclusions in the scientific literature are erroneous, and tests of medical dangers or treatments are often contradictory and confusing.

Replicating a result helps establish its validity more securely, but the common tactic of combining numerous studies into one analysis, while sound in principle, is seldom conducted properly in practice.

Experts in the math of probability and statistics are well aware of these problems and have for decades expressed concern about them in major journals. Over the years, hundreds of published papers have warned that science’s love affair with statistics has spawned countless illegitimate findings. In fact, if you believe what you read in the scientific literature, you shouldn’t believe what you read in the scientific literature.

“There is increasing concern,” declared epidemiologist John Ioannidis in a highly cited 2005 paper in PLoS Medicine, “that in modern research, false findings may be the majority or even the vast majority of published research claims.”

Ioannidis claimed to prove that more than half of published findings are false, but his analysis came under fire for statistical shortcomings of its own. “It may be true, but he didn’t prove it,” says biostatistician Steven Goodman of the Johns Hopkins University School of Public Health. On the other hand, says Goodman, the basic message stands. “There are more false claims made in the medical literature than anybody appreciates,” he says. “There’s no question about that.”

Nobody contends that all of science is wrong, or that it hasn’t compiled an impressive array of truths about the natural world. Still, any single scientific study alone is quite likely to be incorrect, thanks largely to the fact that the standard statistical system for drawing conclusions is, in essence, illogical. “A lot of scientists don’t understand statistics,” says Goodman. “And they don’t understand statistics because the statistics don’t make sense”…

What’s one to make of the stream of “eat this,” “avoid that” studies surfacing nearly daily? It’s an odds-on bet that readers will find out in the complete *Science News* story, “**Odds Are, It’s Wrong**.”

**As we tell Monty that we’ll take what’s behind Door #2**, we might recall that it was on this date in 1905 that Albert Einstein kicked off “Annus Mirabilis” with the publication of the first of his four epoch-making papers in *Annalen der Physik*— this one, proposing energy “quanta”– thus kicking off the year in which he reinvented physics and our understanding of reality.

The second of those papers, on Brownian motion, was the very first work of “statistical physics.”

Einstein, dressed for the patent office, 1905

**Happy Náw-Rúz!** This date in 1844 was the first day of the first year of the Bahai calendar.