## MatheMagician

(Submitted by Skepticality listener and friend of the blog Christopher Brown.

Hi all:

My son, Ethan Brown performs a Mental Mathematics stage show. A few months ago, he developed a new piece for his act. It’s a version of an old presentation puzzle known as The Knight’s Tour.

Traditionally, performers have allowed audience volunteers to select a square on a Chessboard. The performer then begins on that square and theoretically moves a knight around the board using only legal knight moves (which are “L” shaped). The goal is to land on every single square on the board without landing on any square twice.

Ethan adds an additional twist to this trick by allowing the audience to also select the final square on which the knight must land, finishing the puzzle.

Since debuting this new trick, he has had a chance to perform it 5 times. 3 out of those five times, the two audience members selected the exact same two squares (only they were reversed in one of those times). Our back of the envelope calculations place the Mathematical odds at 1 in 107,374,182, though I suspect something else might be going on here. We have video of 2 of the performances if you’d like to see it. Could there be something psychological that causes people to gravitate to these squares much like people often pick “Ace of Spades” when asked to randomly think of a card?

I have attached photos of the three final Knight’s Tours. Note where the numbers 1 and 64 are.

Thanks! Let me know if you have any questions at all.

Below are the extended notes provided by cognitive psychologist and statistician Barbara Drescher for use in Skepticality Episode 244.  Take a look and leave your comments below. Also, please be sure to listen to the podcast for our own hilarious commentary. Also, visit Barbara’s blog.

These notes are a bit dense for the podlet, but maybe you can use the story and just skip most of the math.

+++++

First, let’s assume that the choice of square is completely random in all cases.

We are not particularly interested in the odds that the audience would choose those squares because it’s not the squares themselves that are interesting. It’s the fact that the audience chose the same squares the second time Ethan performed the trick. Therefore, we are given the squares by the first audience and we want to calculate the probability that the second audience would choose those particular squares.

To calculate the odds of choosing those particular squares, we must first note the odds for each, which are pretty easy. The odds of choosing the first square are 1 in 64, or .015625. The odds of choosing the second square are 1 in 32 (since you are limited to only white squares and all white squares are available), or .03125. The odds of choosing both is:

.015625 x .03125 = .00048828125 or 1 in 2,000

1 in 2,000 is the probability that the audience will choose the same squares on the second round that it did on the first round.

The third instance is a bit different because, although the audience chose the same squares, the starting and ending squares are backwards. The calculation is partially the same, but if we allow either square to be the starting square, we are now asking a different question. We now want to know the probability of choosing that specific black square to start and white square to end, or that particular white square to start and black square to end. So, we start with the probability of each scenario, which we know to be about 1 in 2,000, then double it (it is not possible to choose both, so there is no joint probability to subtract). So, the probability of choosing either the same squares or the same squares in reverse on any subsequent game is about .00098 or about 1 in 1,000.

Since each time Ethan performs this trick, there is about 1 in 1,000 chance that the audience will choose those same squares as start/end points, the probability that it would happen on the 3rd, 4th, or 5th time that he performed it is about 3 in 1,000, or .003.

So, taken as a whole, the probability of the audience repeating the first (exact) choices on the second performance and choosing the same squares on one of the three subsequent performances is about .0000015, or 1.5 in a million. So not quite one in a million…

But that is all assuming that the choices were random. I saw nothing in Ethan’s posture or delivery that would suggest any given square as a starting point. However, we do know that human beings don’t do anything at random. I doubt that anyone has conducted studies to determine which squares someone is likely to choose if they are in this particular situation, but I think it is fairly safe to say that they are at least twice as likely to choose squares in the middle of the board than on the edges. I would be interested to find out if that is true, but let’s assume that number is accurate.

That changes the entire game.

We could simply double the probability of choosing those same squares in the second performance, but that wouldn’t give us the whole picture. Now we have to consider the probability of choosing those squares in the first round, because it is no longer a uniform distribution.

If we consider that someone is twice as likely to choose a square that is not on the edge, the probability of choosing that particular square is now .02, or 1 in 50. Likewise, the probability of choosing an ending square that is not on the edge is about 1 in 25. So the probability of choosing those particular beginning and ending squares is:

.02 x .04 = .008 or 1 in 125.

And now the probability of choosing the same squares, with either as the starting square, is about 2 in 125.

And that makes the probability of this scenario about 1 in 31,250.

But I think it is worth noting that the probability of those two squares being chosen at any given performance is independent of the outcome of other performances. It ranges from 1 in 1,000 to 2 in 125, which isn’t exactly “crazy”. But if it keeps happening, I’m going to think seriously about setting up a betting pool.

(Submitted by Skepticality listener Michael Farese.

I have less of a story and more of a question. My girlfriend is from New Jersey and has a very, um, animated personality. While driving, she often gives people certain gestures, honks, flashes headlights, etc.

I always tell her that she needs to be careful and that she shouldn’t do things like that because there are crazy people out there who might try to run her off the road (or worse) in a fit of road rage. She tells me that I’m being ridiculous and that she has a better chance of getting struck by lightning.

My question is: does she have a better chance of getting struck by lightning? Am I worrying about something that has only a negligible statistical chance of occurring?

Looking forward to some insight!

Below are the extended notes provided by cognitive psychologist and statistician Barbara Drescher for use in Skepticality Episode 243.  Take a look and leave your comments below. Also, please be sure to listen to the podcast for our own hilarious commentary. Also, visit Barbara’s blog.

Hmmm…. Well, finding statistics about road rage is difficult, mostly because the definition of “road rage” is fuzzy. However, after looking at several different sources, I believe it’s safe to say that it seriously injures or kills around 1500 people in the U.S. per year, and that doesn’t include incidents in which only minor injuries or property damage are involved.

By contrast, the number of people who are injured by lightning in the U.S. each year is fewer than 300. On average, the number killed is 33.

Several websites echoed this sentiment written in About News:

“Statistics tell us that most all of us have been involved in an aggressive driving experience either as the victim or the aggressor at some point in our lives.”

Yet the lifetime chances of being struck by lightning at some point in one’s life are about 1 in 12,000. So I’d go with the author on this one.

## Watch This Coincidence

(Submitted by reader who prefers to remain anonymous.)

I grew up in Alaska, and didn’t move to Michigan until 1990. After a few years of marriage, my wife and I decided to buy a new house (2002). We just contacted a realtor and looked on the internet ourselves. We finally decided on a house in a town about 10 miles away, because we thought it was a great deal.

I was a construction worker at the time.  About 5 years later, I got into watchmaking, and 4 years after that I opened my own watch repair shop in town.  There were many open storefronts for rent, and we finally decided on the one that looked like it was in the best condition. Soon after, I started researching the local watchmaker who lived and worked in town (he died in 1910). Long story short… not only am I in a storefront literally across the street, but I’m related to both him, and one of the founders of the town.  I don’t think I need to tell you that watchmaking is not a common profession. My great-grandfather was the watchmaker’s second cousin. And I’m a descendant of the brother of one of the founders of the town.

The only thing that makes the story sound less coincidental is if I admit that my mother grew up in another town about 40 miles away, and that her family has lived in this area since the 1830’s.

## Holy Comical Coincidence, Batman!

As I have 2 monitors, I like to watch something on one while surfing the web on the other. I decided to re-watch Batman the Animated Series.

While browsing through stuff in an artist community website, I came across a little fan comic derived from one of the episodes of Batman (coming across comics isn’t rare, but this is the first one I’ve seen derived from a specific episode). I was about to pass it by until I noticed that the comic supposedly took place during the exact episode that I happened to be watching!

## The Spooky Cab Ride

(Submitted by Skepticality listener Celestia Ward

Greetings. I had a strange coincidental experience a couple of decades back that, unfortunately, wasn’t cute or funny. My odds-must-be-crazy story is actually kind of gruesome and not for the weak of heart. So if you don’t mind a change of pace from your typical stories, I’ll tell you mine.

Some years ago, in Baltimore, I worked part-time with a small crew of artists in the tourist district. There were maybe eight of us. After night shifts I would routinely take a cab home; as a young female, waiting for a bus late at night could feel a bit lonely and dangerous. I would walk across the street to the large hotel taxi stand and usually there would be one or two cabs waiting.

One Sunday night I hopped into the one waiting cab and the driver told me he had just gotten paged by one of his “regulars” and would need to go pick her up–but if I wanted to ride along he’d drop me off afterward for a reduced fare. I had never had a driver offer this before, but there were no other cabs at the stand and a cheaper ride sounded good to me. I was in no hurry.

This regular client was a nurse who was just getting off her ER shift at the major hospital in the city center. We chatted as we rode, and she described the victim of grisly crime that had come in the previous night. An eighty-year-old woman had been attacked by her adult son, who lived with her and had a history of mental illness. He had come home from a drinking binge, accused her of stealing his money, and beat her up–even cut into her lips and cheeks, the nurse said, convinced, in his psychotic state, that she was hiding money in her mouth.

The cab driver and I were horrified. She said that the police had this man in custody and were expecting to charge him with murder. The old woman was in very bad condition and not expected to recover.

The nurse was dropped off at her house, then the cab driver took me home at his promised discount, and I just assumed that would be the last I heard of that awful scenario, unless the local news was covering it.

I went to work the next night and saw a couple of coworkers with grim expressions on their faces. They told me that Joe (I am changing his name) wouldn’t be working with us anymore. I first assumed that he’d finally been fired–Joe was kind of a jerk, had some issues and drank too much. No one really liked Joe.

It hit me sideways when my coworkers told me he had been arrested–for killing his mother! Out of the whole city, out of all the times I had taken a cab, I had ended up in the one taxi cab that–unknown to me at the time–got me a firsthand account of a murder committed by a coworker.

Tell me, what are the odds??!

Below are the extended notes provided by cognitive psychologist and statistician Barbara Drescher for use in Skepticality Episode 242.  Take a look and leave your comments below. Also, please be sure to listen to the podcast for our own hilarious commentary. Also, visit Barbara’s blog.

It’s hard to say what the odds are without more information. The population of Baltimore at the time would be helpful, but not entirely, since the odds are increased a great deal by geography–the proximity of Joe’s home to his work place and the hospital where his mother was taken are not coincidental. So, I can say that the odds are much higher than one might think, but it is still quite a coincidence, and similar to stories I have heard before (I even have a similar story myself).

It is a gruesome story, and that gruesomeness enhances the chill and eeriness of the coincidence.

## John’s New TV

(Submitted by TOMBC Team Member John Rael)

The day I went to my bank in order to get a personal loan, I came home, turned on my LCD TV (Westinghouse LVM-47W1), which I’ve owned for six years, and started seeing random ‘snowlike’ pixels on the screen. I turned it off in order to turn it on again… it would not turn on again.

I unplugged it and replugged it. Nothing. It was officially dead. Even though its standby light was on, and it kept making a slightly high pitched hum sound.

Keep in mind, without the loan I had just received (that very day), I would not have been able to afford another television until at least October. Anyways, I’m not sure how relevant any of that is to the coincidence, but there you go. Feel free to incorporate any info you happen to know about me personally (career, lifestyle, etc.). Also, feel free to ask me any questions.

Below are the extended notes for use in Skepticality Episode 241 provided Edward Clint.  Ed Clint produces the Skeptic Ink Network and writes about Evolutionary Psychology, critical thinking and more at his blog Incredulous. He is presently a bioanthropology graduate student at UCLA studying evolutionary psychology.  Take a look and leave your comments below. Also, please be sure to listen to the podcast for our own hilarious commentary.

TV used to be pejoratively called the “boob tube”, until computer monitors became the rightful heir to that meaning, partly because televisions used to be cathode ray tubes. The cathode tubes of our primitive low-def ancestors were electron guns firing away at the screen one pixel at a time. Today’s liquid crystal display (LCD) TV technology is much more reliable, having fewer moving parts, and no electron gun. Thanks to this tubal migration, today’s tube-less TVs can have a mean-time-between-failure of 100,000 hours. This means that, on average, if you watched 5 hours of TV a day, it would take 54 years for the device to fail. A bit less if you like Peter Jackson movies.

TV failure in general is pretty rare. Then again, John, you’re probably not an average user. I’m told you spend a large amount of time and energy on making and consuming videos for the internets and whatever other media outlets still exist. I assume that means you work with lots of footage of cats and people falling off of things. So maybe you really put that Westinghouse through its paces. Even if you used it 24/7, it would probably take 11 years to reach the statistical breaking point.

What’re the odds you’d just happen to be able to replace a broken set on the day it breaks? A fairer question is, how many different expensive things breaking that day could have seemed like a strange coincidence? I have not been to your house, John, but I know you don’t drive, and I will assume it is populated with a variety of large fancy cameras that aren’t compensating for anything, some high end editing equipment, and at least two fancy blenders with way more settings than anyone could possibly need. I’m not sure why I assume there’re blenders, it just feels right. The breakage or loss of any of these items on a given day still isn’t too likely, but the odds are more moderately unhinged than crazy, which seems about right for John Rael.

## A triple play birthday!

Today is my birthday, so here’s a birthday-related anecdote for you.

About a dozen years ago, I went to a lecture at a nearby school.  As we waited for the lecture to start, the lady in the seat to my left started talking with me.  After a little while, she mentioned her birthday is August 11th.  The lady in the seat to my right overheard, and she told us that HER birthday is also August 11th.  At that point, I revealed that my own birthday is August 11th, too!

None of us had ever met or even seen each other before, but we’d all just happened to sit side by side by side!  Since then, the lady on my left has become a good friend, and every year we celebrate our birthdays together.

## Gamer’s Timed Coincidence

(Submitted by Skepticality listener Lee Christie)

Hi, I listen to your segment on Skepticality and encountered a coincidence today that I felt would be fun to share with you.

After watching a show on TV, I began playing “NES Remix” for Wii U (a new downloadable game which gives you hundreds of mini challenges lasting usually about 10-20 seconds each from 16 classic Nintendo games from the mid ’80s).

I was playing using the portable gamepad screen alone, and left the TV on the same channel I was no longer watching. It was playing a program called “Rude(ish) Tube” with an assortment of amusing clips and at that moment, a series of clips involving cats.

I had been playing the challenge stages of “Donkey Kong Jr.” for a while and then just as I switched to playing the first underground stage challenge of “Super Mario Bros.” remix (which has a 10 second timer on the challenge to collect 4 coins then ends), the TV clip show started showing a clip (lasting about 15 seconds) of an cute ginger cat jumping around, reacting to the unmistakable music and sound effects from an underground level of “Super Mario Bros.”

I don’t recall another instance of hearing the underground-level SMB theme on a TV show, and certainly not coinciding with me playing 10-second underground-level challenge of SMB.

Note: as “NES Remix” was only released 6 days ago and I assume these shows have a longer time between recording and air, I suspect the people who submitted the clip of the cat were playing the original 1985 “Super Mario Bros.” or a re-release of it, not the recent “NES Remix” as I was playing.

## That’s Textbook

(Submitted by Skepticality listener Rich Catalano)

I am an English teacher in Japan. I have used a variety of ESL textbooks over the years, but this year caused a stir. Why?

In one of the stock images, I appear in the background. I checked, and this particular photograph was from Getty Images, a famous stock-image company. The setting is a museum, and it shows two people looking at an unseen piece. I am in the background, alone, looking at a different piece. It is obvious that I am not the focus of the photograph, so perhaps I was captured inadvertently.

I showed this photo to everyone who knows me, including my ever-doubting wife, and they all concur that the image is me.

In the past, I went to Europe every summer with a group of students and always visited museums. Perhaps this is when the photo was taken.

Not sure if the odds are against this, but they seem to be.

Below are the extended notes for use in Skepticality Episode 240 provided Edward Clint.  Clint produces the Skeptic Ink Network and writes about Evolutionary Psychology, critical thinking and more at his blog Incredulous. He is presently a bioanthropology graduate student at UCLA studying evolutionary psychology.  Take a look and leave your comments below. Also, please be sure to listen to the podcast for our own hilarious commentary.

Rarely can someone say that they have a “background in teaching ESL” and mean it so literally. The odds must be astronomical. No, probably worse than that, because the odds of liking astronomy are pretty good. Who doesn’t want to tool around the universe in a giant chrome leech with Neil deGrasse Tyson? When scientifically analyzing likelihood in questions such as this, we separate the larger question into two smaller ones termed the “boring part” and the “interesting part”. It’s a Bayesian method. Probably. Anyway, the comparatively boring part is how likely is it you’d wind up in a stock photo in the Getty Images library? Or, not just Getty but any similar image supplier? For a world-trotting agorafile like you, maybe not as bad as you think. The top eight such services have a combined 155 million images. There is a constant demand for new images, and every day thousands of these get sold to hundreds of clients from cable news networks to product catalogs. Not all of those 155 million images are photos of people, but your odds of being in there are higher if you are a in a labcoat holding a clipboard, like to stand in front of a lot of sunsets, or, in this case, were looking at one of those dumb Louvre statues that doesn’t even have arms.

The more interesting part is, what are the chances that image would find you once it was in the Getty bank, and in a textbook you’re teaching from? Unless you’ve been doing this job for 60 years, probably on the low side. On the other hand, our increasingly visual media-rich global culture might be making this sort of thing more common. Just two weeks ago The Nation website reported a math textbook in Thailand had to have its cover changed because the bespectacled professional woman center frame is Japanese adult cinema actress Mana Aoki. You and Ms. Aoki have something in common: your images might have been sold or used dozens or hundreds of times by now. Plus you’re both apparently highly recognizable by a small set of Japanese people. That’s a feather in your cap.

## Coincidence? I think so!

(Submitted by Skepticality listener Michael O’Dea

Hi there,

I enjoy the show and want to tell you my against-the-odds-story.

I am from Dublin, Ireland and I was on vacation in Boston, visiting my cousin about 20 years ago.

There was a free public concert in the Boston Common park. (It was Kid Creole and the Coconuts, not that that is relevant!)

I was with an American friend who was a server in a Boston restaurant (Legal’s) at the time. As we enjoyed the music he met a colleague from the restaurant who was with a companion and they chatted for a few minutes as we watched the gig. My friend then went to introduce me, when the companion turned around it was my next-door-neighbour from Dublin!

We had not seen each other for years and had no other connection of any kind other than growing up in adjacent houses.

What do you think?

Below are the extended notes provided by cognitive psychologist and statistician Barbara Drescher for use in Skepticality Episode 239.  Take a look and leave your comments below. Also, please be sure to listen to the podcast for our own hilarious commentary. Also, visit Barbara’s blog.

I think this is an interesting coincidence! Normally, I would talk about the factors that would increase the probability of this happening, so I will, but there are really very few. People living next door to one another are much, much more alike than two people chosen at random from the global population. They are more likely to be close together in S.E.S. (socioeconomic status), for example. They are more likely to be exposed to similar cultural icons (such as music genres). Factors such as these may exponentially increase the probability of running into each other at just such an event.

However, given the astronomically small base probability (e.g., given all of the people in the world, the probability of any two people, chosen at random, would meet), this is still a story with crazy odds.

Consider the factors that don’t really come into play here, but have in similar stories we have encountered. For example it is unlikely both been inspired to visit by the same event (e.g., hearing a mutual friend talk about visiting Boston). They may have been inspired to visit (assuming the companion was also visiting and not living there) by cheap airfare to the U.S., but then why choose Boston? The probability that they all met each other through mutual friends is greatly reduced by the fact that the Americans know each other because they work together (unless, of course, they knew each other before working there).

So we must rely on the mathematical rule that we should expect at least some low- and even astronomical-probability events to occur in our lives, given the large number of events that occur.