Tag Archives: probability

Things 106: Best at Chess, Art of Science, Crowdfunding

Tim Link – Competitive Sandwich Making
Last week Clare and I ran a game based on tessellating pieces of cheese to make the best sandwich for the Hide&Seek Sandpit event. You can read about it and see the photos on my project blog, Tower of the Octopus.

Puzzle
In a chess tournament in which anyone can use any means available to them to come up with their moves, who would win? Some possible answers to give an idea of what kind of thing I’m talking about here:

  • A high-ranking chess Grandmaster
  • A really good chess-playing supercomputer
  • A huge team of moderately skilled players with some method of combining their ideas
  • A moderately skilled player with access to a moderately good computer that can run some basic chess calculations

(I had wondered about this in an abstract way before, but recently found out that has actually been done. I’ll relate what happened in that event next week, but you could of course try to Google as well as guess the answer if you wanted).

Video
(Via Phil): Art out of Science:
(Two views of the same thing. If your browser is up to it, you could try watching both videos simultaneously – start the bottom one 20s after the top):

Links
Kickstarter is one of my favourite things on the internet: people with an idea for something get a platform from which to shout about it, and to collect pre-orders or donations from people that like the idea. If there’s enough interest, the project can go ahead, and everybody wins.

So far I’ve helped fund two comics, the Wormworld Saga app (which saw so much success the creator, Daniel Lieske, decided he could give up his day job), and I’m currently backing The Endangered Alphabets Project, which is the kind of thing I like to imagine in a vague way is going on in the world, but I now have the opportunity to facilitate it directly (also, it’s only just on track to hit target, so do go check it out).

You can follow Kickstarter on Twitter, or go to their home page and scroll to the bottom to sign up for the weekly newsletter which highlights the most interesting projects.

IndieGoGo is similar but for reasons I can’t really pin down doesn’t work as well for me.

Crowdfunder is a UK version which I don’t tend to find as inspiring, but would probably be the best one for someone in the UK to create a project with (since Kickstarter requires a US bank account).

Quote
Overheard in the maths common room when I was studying for my PhD at Royal Holloway:

But nobody knows what probability is! Probability is defined in terms of randomness, and randomness is defined in terms of probability!

Answers to Monty Hall and the Two Envelopes
Last week I asked about the Monty Hall problem, which I should have introduced before the Two Envelopes problem I set two weeks ago.

The Monty Hall problem has a nice Wikipedia page, the most helpful part of which is probably the decision tree showing all possible outcomes.

In brief, the answer is that you should switch after Monty shows you an incorrect door, but certain misguided instincts steer most people away from that choice. The Endowment Effect and Loss Aversion mean that regardless of probability, people fear they would regret “giving up” their first choice more than sticking with it if they end up losing.

The more subtle effect is an instinctive (or partially trained?) feeling that the choices of others have no effect on the probabilities of our own choices in these kind of contexts. This is true when the other person has just as much information as you, but that is not the case here – Monty knows where the car is, and uses that information to ensure that he always opens a door with a goat behind it. So he has more information than you, and when you see his choice you gain some information.

Or to give an answer that might go with the grain of instinct for some people, consider this: there is a 2/3 chance that the car is behind one of the doors you don’t pick. Monty shows you that it definitely isn’t behind one of them. So there’s still a 2/3 chance the car is behind the other one, and a 1/3 chance it’s behind the one you first chose.

As for the Two Envelopes, it turns out this is more difficult than I originally remembered. Again, there’s a great Wikipedia page on the subject, which has quite a lot of detail.

As Thomas noted, a key phrase missing from the subtly specious argument for swapping is “Without Loss of Generality” (WLOG), which one must always be careful to check whenever substituting a variable (in this case, the amount in the envelope) with a specific figure (£10 in the example I gave).

Is it true that the reasoning I gave based on having £10 in the envelope truly retains the generality of the problem – would the reasoning also hold for any other amount? In short, no. For example, there could be 1p in the envelope, or any odd number of pence, in which case we would have to conclude that we had the lesser envelope (although this still means you should swap). More dramatically (here we imagine the envelopes contain cheques, and that these cheques are totally reliable), your envelope could theoretically contain over half of all the money in the world, in which case you can be sure the other envelope contains less. More realistically, it could contain more than 1/3 the amount of money you expect the person filling the envelopes to be willing to give away, in which case you would strongly suspect the other envelope to contain the lesser amount.

If you’re interested, do read the full Wikipedia article, and meanwhile, remember to watch out for unjustifiable WLOGs.

Things 105: Pervasive Game Event, Monty Hall, American Politics

Upcoming Event, Thursday August 4th
People that like Things are very likely to like this. On Thursday August 4th from 7pm-10pm, Hide & Seek are running a Sandpit gaming event at the Southbank Centre. In practice this means you get to turn up and play lots of interesting games (for adults) for free. Having been to quite a few of these in the past, I highly recommend it as the games are always fascinating and inspiring. I’m particularly excited about this one because Clare and I will be running a game ourselves, one based on the age-old problem of tessellating pieces of cheese to make a perfect sandwich.

More details of the event can be found here, and the official Facebook event is here. Let me know if you think you can make it!

Puzzle – Monty Hall
After talking to some people about last week’s Two Envelopes puzzle, I realised that many Things readers may not be familiar with the Monty Hall Problem, which one should really understand before tackling the Two Envelopes. So I’ll state that here, then go through the answers to both in the following Things.

In the Monty Hall problem, you are in a gameshow presented by the eponymous Monty. You are asked to choose one of three closed doors. Behind one of the doors is a nice car that you apparently want to win. Behind the other two doors are goats. If you choose the door with the car behind it, you win the car. If you choose a door with a goat behind it, I don’t think you win a goat, but you definitely don’t win the car. Basically the goats are there just for comedic effect.

So you choose a door, pretty much at random. At this point Monty (who knows where the car is) opens one of the remaining two doors to reveal a goat. He does this in every episode of the show – whichever door the contestant chooses, Monty will always then open one of the remaining doors to reveal a goat. He then offers you the chance to switch from your first choice to the other unopened door. The question is: should you switch?

Link
Sometimes The Onion packs a headline with so much satire it barely needs the accompanying article. Most recently I was impressed by American People Hire High-Powered Lobbyist To Push Interests In Congress.

Quote
From Jon Stewart’s speech at the “Rally to Restore Santiy”:

The press can hold its magnifying up to our problems bringing them into focus, illuminating issues heretofore unseen or they can use that magnifying glass to light ants on fire and then perhaps host a week of shows on the sudden, unexpected dangerous flaming ant epidemic.

Picture
This National Geographic healthcare data visualisation achieves a rare feat: showing the data in an unconventional way that nonetheless actually tells a story with the data quite well. Charlie Park has some great commentary on why a scatter plot of this data isn’t actually as useful in his general discussion of slopegraphs.

Click for big:

Things 104: Power of Music, Misleading Impressions, Two Envelopes

Video
The effect of music on the brain is a very interesting thing that varies tremendously by individual. Last year I discovered a track that has an incredibly powerful mood-altering effect on me: Olympians, by a band with a potentially offensive name. It took a couple of initial slightly bemused listens before it properly seeped into my brain, but now as soon as I hear this track, I feel unbelievably positive, and become filled with an absurd confidence.

Unfortunately I suspect the fact that this track is so resonant for me also suggests that it’s very specific, and it will seem really quite boring to most others. But I find it so amazing I just have to share it anyway. So first, here’s a short version with a video to slightly entertain you while you wonder what on earth I’m going on about:

And if you are so inclined, here’s the full length version:

Tim Link
I saw The Lion King in 3D at Edinburgh International Film Festival, and reviewed it here. The short version of my review would essentially be this:

Quote special: Misleading Impressions
Thanks to Last.fm recommendations I discovered Brian Transeau (BT)’s album This Binary Universe, which turns out to be a bit different to his other albums. As I listened to his back-catalogue I thought I detected an incredible sense of optimisim and positivity. When I later found Brian Transeau was on Twitter, I found this impression was entirely correct. Sample tweet:

5am and time for our first ever sunrise, father daughter bike ride. Today is already #WIN Good Morning!

My favourite musician is probably Jon Hopkins who I now listen to instead of any other Chill Out music since for me he somehow trumps pretty much the entire genre. He is behind some of the most relaxing and beautiful tracks I know, so I was curious to see what he was like on Twitter. The answer: actually a bit different. Brilliantly, this was the first Tweet of his that I read:

I wish one of James May’s Big Ideas was to FUCK OFF

Finally, moving away from music, I referenced Mitch Hedberg’s famous escalator line in my Lion King 3D review:

An escalator can never break – it can only become stairs.

Realising I was unfamiliar with his work, I ended up reading through his Wikiquote page, and found much to like, such as:

My belt holds up my pants and my pants have belt loops that hold up the belt. What the fuck’s really goin on down there? Who is the real hero?

and:

When you go to a restaurant on the weekends and it’s busy they start a waiting list. They start calling out names, they say “Dufresne, party of two. Dufresne, party of two.” And if no one answers they’ll say their name again. “Dufresne, party of two, Dufresne, party of two.” But then if no one answers they’ll just go right on to the next name. “Bush, party of three.” Yeah, what happened to the Dufresnes? No one seems to give a shit. Who can eat at a time like this? People are missing! You fuckers are selfish. The Dufresnes are in someone’s trunk right now, with duct tape over their mouths. And they’re hungry. That’s a double whammy. Bush, search party of three, you can eat when you find the Dufresnes.

So after that I naturally looked him up on YouTube, and at that point discovered him to be completely different to what I had imagined:

http://www.youtube.com/watch?v=2491LucLa1g

Gone, try this one:

Picture
A lot of infographics annoy me, but I like the idea of bringing together the data that drives this one so much I don’t mind its shortcomings.

Puzzle – The Two Envelopes
I can’t believe I haven’t put this one in Things before.

In a standard abstract setting with no distracting details, you and another person are presented with two envelopes. One envelope contains some money (but you don’t know how much). The other envelope contains twice as much money. You get to select an envelope, and you get to keep however much money is in it. The other person gets the other envelope. There isn’t anything to go on, so you choose one of the envelopes for arbitrary reasons.

Before you get to open it, you are offered the chance to change your mind, with the following reasoning:

You don’t know how much money is in your chosen envelope, but for the sake of argument let’s say it’s £10. That means you either have the envelope with twice as much money (so the other contains £5) or you don’t (so the other must contain £20). So if you decide to swap, there’s a 50% chance you get that £5, and a 50% chance you end up with the £20. Since you currently have £10, that means there’s a 50% chance of effectively losing £5 and a 50% chance of gaining another £10. Imagine if the universe split into two at the moment you made that decision – one of you loses £5, the other gains £10, so on average you gain (£10 + (-£5) )/2 = £2.50. Since the average gain is positive, clearly that’s a gamble worth taking, and you should definitely swap.

This is of course a strange conclusion. You effectively chose an envelope at random, so how does swapping it improve your odds of getting more money? The paradox is even more stark if we consider the fact that the other person could be convinced to swap by exactly the same argument.

Previous Puzzle – Co-operating with yourself
Last time I asked how well you would get on with yourself.

Xuan said:

They say that people you dislike/hate are likely to be people who’s characteristics are most like yours. People are most critical of what they see in the mirror. My clone better not have the same taste in clothes.

Which reminded me of a problem the sci-fi stories don’t tend to go into – if there’s suddenly two of you, you’re going to need some more clothes, and one of you will probably have to find another job, and probably somewhere else to live. Marriages get complicated. Phil suggested David Gerrold’s time-travel sci fi story The Man Who Folded Himself for an in-depth dissection of this kind of problem.

Richard observed that he tends to like people with whom he shares attractive personality traits, and dislike those that share his negative personality traits, suggesting that the latter may be because they serve as a reminder of these aspects of himself. This potentially makes the question even harder to answer, although one might guess that a negative would trump a positive and ultimately lead to the kind of confrontations that usually crop up in sci-fi versions of this problem (and endorsing Xuan’s observation).

I think the question raised by The Man Who Folded Himself of co-operating with a version of yourself in the future is a clue to how we can actually ask this question of ourselves. In a very real sense, we really do choose how much to co-operate with our future selves every day: will you do a chore now, or will you force your future self to do it instead? Will you eat all of the cake, or will you save some for your future self? If you know how you generally answer those questions, I suggest this gives you an idea of how well you would get on with yourself.

In practical terms, just thinking of these kinds of questions in this framework makes me more likely to co-operate with my future self, which is probably a good thing. Well, I’m glad that my past self thinks that way, anyway.

Things 9: Statistics Art, Dramatic Prairie Dog, Flag Diagrams

(Originally sent April 2008)

It’s the return of Things!

This week’s film – one line review:
Last night I saw Son of Rambow, which was kind of Beano-like, over the top, strange, and put together with more heart than writing competence much like the film-within-a-film it portrays, and fortunately heart is the most important aspect of a film, so that was rather good.

Next week’s films:
Next week I plan to catch the preview of Happy-Go-Lucky. It’s a film about the power of optimism by Mike Leigh, and Mark Kermode was enthusing about it, so it sounds well worth seeing.

Imdb rating: 8.6 /10 (from 87 votes)
Rotten Tomatoes rating:  N/A (insufficient data)
[Now gets 92% on RT – metatim, 16/5/10]
Clip : http://www.youtube.com/watch?v=aqTlFY84yYU
[Link is dead, I think this is the same one though – metatim, 16/5/10]

The clip doesn’t seem terribly demonstrative given what Kermode was going on about, but it’s the best I could find.

A Puzzle:
In the last Things I left you with the rainbow paradox. I don’t have an answer but I saw a diagram once of the colours the human eye perceives separated by how distinct we perceive them to be, that looked like a clue. If I find it then I’ll use it as the image attachment of a Things one day. [That diagram can be found here, from this Wikipedia article on colour vision – metatim, 16/5/10]

This week: If there are 3 people in a room, what is the probability that at least 2 of them are of the same sex?

A Quote:
Federico García Lorca: The iguana will bite he who does not dream.

A Link:
Chris Jordan has a series called ‘running the numbers’, in which he takes amazing statistics and makes art out of them:

http://www.chrisjordan.com/

A video:
This 5-second video was big in June 2007 but I somehow missed it. It’s an internet classic and it is vital that you see it with sound:

httpv://www.youtube.com/watch?v=jHjFxJVeCQs

A picture:
Below is an image which, like Chris Jordan’s art, also presents statistics in an interesting way.