Probabilities and Free Toys, Part 2

Last time, I looked at solving a probability question:  for a set of 3, 4, 5 or n toys which are given away free inside a cereal packet, what's the probability of obtaining the full set of toys after buying the same number of packets (e.g. for 5 toys, getting all 5 after buying 5 packs).  

This time, having solved the easier question, let's take a look at the harder question:  with 10 toys (or a number larger than three or four), how many packs do I have to buy to be 50%, 70% or 90% sure of having the full set?

Once again, let's start with two toys (start small!):

After buying two packs, the probability of success is 0.5.  The two successful combinations are AB and BA, and the two unsuccessful are AA and BB (i.e. I get the same toy twice).


After buying three packs, the probability of success rises to 6/8, which is 0.75 or 75%.
There are eight total combinations (from AAA, AAB through to BBA and BBB), but only two are unsuccessful (AAA and BBB are unsuccessful), leaving six successful combinations.


After buying four packs, the probability of success rises to 7/8.  There are 16 total combinations, but the number of unsuccessful combinations remains at two, and the number of successful combinations rises to 14.


Generally, for the two-toy problem, the probability of getting a successful combination after t turns is equal to (2^t - 2) / 2^t  or, in words, the total number of combinations minus unsuccessful, divided by the total number of combinations.


This, however, is where it gets tricky.  With three toys and three packs, there are just six successful combinations (the exact permutations, in alphabetical order, are ABC, ACB, BAC, BCA, CAB and CBA) and 27 total combinations (which confirms what I proved above for the three toys and three packs case).  With three toys and four packs, the number of successes rises to 36, and the total number combinations is 81.  The success probability is 36/81 which is 4/9, coincidentally double the probability after three packs.  After five packs, the number of combinations rises to 243.


Looking at the number of combinations I've seen so far, for the three-toy problem, it's risen from 9 to 81 to 243, rising by a factor of 3 each time, and n=3, the number of toys squared. In fact the denominator, the total number of permutations, is simply n^t (number of toys raised to the power of the number of turns or packs).  This is also known as the formula for 'permutations with repitition'.


Now I need to identify the number of successful permutations - those that contain one of each of the toys.
For n=3 toys and t=3 turns, there are 6 successful permutations (as listed above).


For n=3 toys and t=4 turns, there are 36 successful permutations


Now, the question becomes - how many will there be after five turns?


Let's go back to four turns and look closely at each of the permutations we have.  We have the 36 successful ones, ABCA, ABCB, AABC and so on.  For these 36 successes, it doesn't matter what we pick next, we'll still have a successful combination; there are three options for each of these, so that's 36x3=108 successes for each of them.


There are also the 3 combinations AAAA, BBBB and CCCC which will not produce a success, irrespective of what we pick next, so that's 3x3=9 fails.


This leaves the rest, which logically must contain two different toys.  We've covered the ones which already contain three different toys, and we've looked at the ones which contain only one toy.  Since there were 81 combinations in total after four goes, there must be 81-(36+9) = 36 combinations which contain two different toys.  There is a probability of 1/3 of the next choice being the correct one, which equals 36 x 1/3 = 12.


So, after five turns, we have 108 + 12 = 120 successful permutations.  
Let's review:


After three turns: 6 successful permutations
After four turns: 36 successful permutations
After five turns:  120 successful permutations


Let's take a look at six turns, based on the process we used for five turns.

After five turns, we have 120 successes which will each yield three more successes, so 3x120 = 360
We also still have the combinations which have just one toy in - AAAAA, BBBBB and CCCCC, and these will each produce three more unsuccessful combinations, irrespective of the next choice.  3x3=9 unsuccessful combinations.


There are 3^5 total combinations after five turns, 243 in total, which means that we have 243-(120+9) = 114 other combinations which have two toys.  A third of these will become successful with the sixth turn, which is 114/3 = 38.


So, after six turns, we have 360 + 38 = 398 successful permutations.  I've deduced the formula for working out successful permutations in an iterative manner, but I don't have the computing power to determine the 10th, 15th or 115th term without knowing each one before.  Furthermore, this method won't easily expand to cover five, six, or ten toys. It's all about knowing how many successes you've had previously, and how many certainly won't become successful (because they have n-2 different toys and will require at least two more goes to become successful).


Three toys is an easy case - you either have a successful combination, a combination with only one toy repeated, or a two-toy combination with a 1/3 chance of becoming successful.  With four toys, you may already have one, two or three different toys, or be successful, and I don't quite see how to sort all that out.

So, next time, it's on to spreadsheet modelling.  I'm going to write a macro that simulates buying the cereal packets and examining the toys, and determining whether or not the combination is a success.  If maths fails, use sampling!

The Probabilities and Free Toys Series

Part 1:  Solving for "What's the probability of getting 10 toys in just 10 packs?"
Part 2:  Solving for "How many packs do I need to buy to be 50%, 70%, 90% sure of getting all the toys?" 

Part 3: Spreadsheet modelling the toys and probability question
Part 4: An elegant solution to 'what's the average number of boxes you'd need to open to get all the toys'?

A Beginner's Social Media Strategy

I say 'for beginners' as I don't feel particularly qualified to discuss it in much detail (as you'll soon see), and this is more an explanation of my background and experience so far.


A few years ago, I set up my own website; you may have seen me refer to it previously. It was set up entirely as an exercise in website-building and tagging - it's all hard-coded HTML. It's tagged with Google Analytics, and I've been monitoring traffic to it since then, doing a little SEO and making changes (and hopefully improvements) to the content here and there. As analysts, we're usually charged with analysing, understanding and reporting stats, and then generating recommendations from them; we're not usually given a logon to a CMS and given free run of a website. By building my own website, I got to play both sides (and realised how time-consuming content generation and site maintenance can be). Better still, I've even been running A/B/C tests on it, and finding out how easy it is to set up (providing you've got multiple content ready to serve).


Anyway, with time, I moved from focussing on the website to a blog. Blogs are, for this JavaScript beginner, much easier than HTML websites, especially when I can't use server-side includes. So, even with a WYSIWYG HTML editor, blog posts are much easier than HTML pages. Once again I found out how to tag my blog by putting javascript includes in my posts and inserting GA tags in my posts, and I've monitored the traffic. I also discovered how many analysts there are out there who are reading this blog (it's not a huge number, but compared to the single digits I was experiencing before I started writing about web analytics, it's a significant uplift). The blog has an 'about me' page which includes my Facebook and Twitter details, so that people can follow me, and I post updates about my blog on Twitter or Facebook or both, and occasionally on the Yahoo Web Analytics forum.


All of which means that I've built up a cyclical path between my social media accounts (where you can find links to my blog) and my blog (which explains how to follow me on social media).


That's not a strategy, that's just a circle. Which brings me to one key question: What am I actually trying to achieve with all this online presence?


Am I trying to get Twitter followers? Am I doing this for my ego, or for PR, or something like that? Maybe, but probably not.  Am I trying to get Facebook friends? No - I've got enough friends (and I've met 99% of them in person) and I've successfully tracked down my best friends from primary school, high school and university. Am I trying to get more people to read my blog? Now then - that seems more likely. What I'm trying to do is to share what I know about various subjects (maths, chemistry, chess, web analytics) and hopefully build up an online reputation as a reliable source of useful, accurate information - to be regarded as a specialist in my field (and possibly even, one day, an expert).


People aren't going to get that level of knowledge about me from my Twitter feed (which, even with my best intentions, is very clouded up with links to miscellaneous stuff I find interesting). They are most certainly not going to get that from my Facebook updates, which are much more personal and include family updates, photos from days out and the like, and are very much about my views and opinions and general chatter. Hopefully, though, my peers and friends will read my blog, where I write my more considered opinions and views, and share what I hope will be useful insights into areas that specifically interest me - as I said, Chess, maths, chemistry and web analytics. The blog also has a Google Analytics goal set up - visitor views the About Me page - and this means that my social media strategy not only has an aim (to get people to read the blog) but a specific goal (find out more about me and my professional skills and experience).


I could go on and build KPIs about blog traffic levels and so on, and on to Twitter followers (excluding the spam accounts) but these will be secondary to getting people to view my profile page on my blog. I can THEN use analytics to tell me which blog post they read before reading my profile, and also where they came from... and then write more blogs on similar topics and post links on similar sources.


And that, in a nutshell, is my social media strategy. I can't say that it's scalable to a company level, but I think the main points (which are probably covered elsewhere) are:


What am I actually trying to achieve with all this online presence (answer in English words)?
What is my actual aim?
What does this look like as an online event (in page terms)? Make this an online 'goal' or 'event' in analytics package.
What type of visitor carries out a success event? Which social media site did they come from?
How do I get more of them? What sort of content do they look at?


I'm not sure if this is a social media strategy, or just a reiteration of a normal online strategy. Like I said, I'm a beginner on social media strategies (despite having a blog, Facebook and Twitter accounts for years) so I'm open to other suggestions!

Probabilities and Free Toys, Part 1

Once upon a time, a long time ago, my high-school maths teacher set an extension problem: I never got chance to tackle it, and I've never tried to since.  I've remembered it through the years, as a friend of mine was able to solve it elegantly, and I never saw his answer.  So, it's time for some closure again.

Here's the question:  

A certain breakfast cereal manufacturer is giving away a free toy inside each pack.  There are ten toys in the series, and I'd like to collect them all.  However, I can't tell which toy I'm going to get when I buy the pack.  The original question was:  what's the probability of getting all ten toys after opening ten packs?  And the follow-up question I'd like to look at is:  assuming that each toy is distributed equally, how many packs would I have to buy to be 50%, 70% or 90% sure of having all ten toys?

Now, bearing in mind that this is a high-school maths problem, it shouldn't take any advanced maths to solve the first question.  The follow-up question is one of my own, and could take me anywhere.

So, let's look at the first question, and let's start with two toys and build up to ten.

After buying two packs, the probability of success is 0.5.  The two successful combinations are AB and BA, and the two unsuccessful are AA and BB (i.e. I get the same toy twice).  But let's look at that as a step-by-step process.  When I buy the first pack, I am certain of getting a toy I haven't got before.  There are two alternatives, A and B, and two successes (either of them).  So the probability is 2/2.  The probability of getting a successful toy with my second pack is 1/2.  There's now only one successful toy (the one I haven't got), but there are two toys available.  To calculate the probability of getting the first toy and the second toy in two packs is 1/2 x 2/2 = 1/2.


This can be expanded to three toys, A, B and C:
Probability of success with first pack is 3/3  (any of the toys is a success)
Probability of success with the second pack is 2/3 (I need to avoid getting a duplicate, so there are now only two successes.  If I have A, then I only need B or C).
Probability of success with the third pack is 1/3 (I now need one specific toy as I have the other two).


So, the probability of success after three packs = 3/3 x 2/3 x 1/3 = 6/27 = 2/9 = 22%


I'll do the case for four toys, before moving to a general expression:
p(success with first pack) = 4/4 as any of the four toys is a success
p(success with second pack) = 3/4 as I already have one toy, and need one of the other three
p(success with third pack) = 2/4 as I have two toys and only two are now successes
p(success with fourth pack) = 1/4 as I only need one of the four toys to complete my set.


So, probability of success with four toys and four packs =
4/4 x 3/4 x 2/4 x 1/4 = 24/256 = 3/32 = 9.375%


There's a clear pattern developing.  For five toys, the numerators will be 5, 4, 3, 2, 1 and the denominators will be 5, 5, 5, 5, 5.  The numerators are multiplied together, 5x4x3x2x1 which is called 5 factorial, and written 5! while the denominators are 5x5x5x5x5 which is 5^5.  Looking back, the same rule applies to four toys, three toys and two toys, and will apply going upwards.

So, the probability of getting all n toys with n packs is n! / n^n

n! is an expression that increases very quickly with n (1, 2, 6, 24, 120, 720, 5040 and so on) but the denominator n^n increases even more quickly (1, 4, 27, 256, 3125, 46656 and so on).  The table below shows n, n!, n^n and the ratio n! / n^n which is the probability of getting n toys with n packs.  For the original question - what's the probability of getting 10 toys in 10 packs, the answer is 10! / 10^10 which is 0.036% (less than one in a thousand).


Next time, I'll look at the harder question:  with 10 toys (or a number larger than three or four), how many packs do I have to buy to be 50%, 70% or 90% sure of having the full set?

The Probabilities and Free Toys Series

Part 1:  Solving for "What's the probability of getting 10 toys in just 10 packs?"
Part 2:  Solving for "How many packs do I need to buy to be 50%, 70%, 90% sure of getting all the toys?" 

Part 3: Spreadsheet modelling the toys and probability question
Part 4: An elegant solution to 'what's the average number of boxes you'd need to open to get all the toys'?

My X Factor predictions for 2011

Now, I reckon I'm a fairly optimistic person.  I look for the best in people and in situations, as a rule, and I will try to give people the benefit of the doubt.  However, I can also be quite cynical.  I don't see it as cynical, other people do, I see it as identifying trends and patterns and expecting them to be repeated - even though I hope for the best.


There is, however, one area where I am just plain cynical - or, alternatively, very good at spotting patterns and trends - and that is with the Saturday evening television monstrosity known as the X-Factor (which I do call the X-Factory given its aim of mass production of pop music and cardboard cut-out pop stars).


Here, then, based on previous years' viewing (despite myself) are my predictions for what we can expect from the Simon Cowell juggernaut this year.


*  At least one finalist to have estranged parent or sibling - I appreciate I'm late with this, given that immediately after the first episode, one of the judges discovered a brother she never knew she had.


*  Gary Barlow to have one of his Take That mates at the judges' house stage (and it won't be Robbie)
*  One of finalists to have been bullied at school
*  There will be the formation of a boy group and girl group, made up of the boy dregs and girl dregs at the end of the boot camp stage.  "We want to put you together into a group [because we haven't got enough groups already]."
*  These synthetic dregs-groups to go through to the live shows (you didn't think the judges would put them together and not let them go through, did you?).
*  These synthetic groups to get eliminated in first two weeks - first the girls (who will dress inappropriately) and then the boys (who can't sing as a team)
*  Simon Cowell to make a guest appearance, to much fanfare and flashing lights
*  Last year's winner (whoever that was) to release album just in time for Christmas
*  Louis Walsh to pick a wildcard act (or just a wild act) which is no good, but which secures the votes of those who deliberately vote for the worst (Jedward, Wagner).
*  There will be extensive media coverage of an apparent spat between two of the judges, probably the two ladies, but possibly the two blokes
*  One of the acts to suffer with a cough/cold/laryngitis/glandular fever part way through the TV shows
*  Two of the acts to form a 'secret' relationship, again with much media coverage


Print out the list, and tick them off.  If there any left by Christmas, I think I'll genuinely be surprised.  In fact, give me a week or two, and I'll probably have some more predictions.

Film review: Cloverfield

Cloverfield

I've been considering add this to my Lovefilm list, although I was never completely convinced.  The trailer for the film made it look a lot like another Godzilla movie - and yes, I've enjoyed the Jurassic Park series, and I've watched bits of the new Godzilla film, but the genre has never really appealed to me.  However, when Cloverfield showed up on the TV listings, I set the recorder and figured I'd pick it up when there was nothing on the TV.  

Cloverfield goes for the innovative approach of filming everything from the first person, with a hand-held camcorder (or at least making it look that way).  The film starts steadily, as one night a huge monster starts ripping up a city.  Which city?  I'll give you a clue:  it's the main city in Independence Day, The Day The Earth Stood Still, Spiderman...  yes, once again, New York is the shortcut for 'any main Earth city'.  Normally, I'd include a summary of the plot in my review, but I'm going to struggle for this film.  There isn't much plot.  Monster rips up city; army arrives eventually; conventional weapons are utterly useless; government authorises use of nuclear weapons.  The remaining story revolves around a group of civilians (including our cameraman) who aren't intelligent enough to run away, and instead, insist on 'documenting' everything.  Their motivation for this isn't clear, and as I watched our characters head towards the danger, while crowds of intelligent people started felling, I lost any interest or sympathy for them.  

The other main character - the monster - had no character at all.  It was extremely difficult to feel anything for it - was it an evil enemy bent on destruction and conquest?  Was it from outer space or under the ground?  Was it - as one of the characters suggested - from under the sea?  Was it lost?  Was this a misunderstood first contact going badly wrong?  It was part - and an unfortunate part - of the film's set-up that it's almost entirely filmed from one perspective; perhaps the film was going for the idea of portraying the details of a monster attack (I refuse to call it an alien invasion) from an individual's point-of-view.  If it was, then it didn't work, for three key reasons:  the cameraman and his associates were not particularly well drawn as characters (despite the occasional flashbacks); they kept running towards trouble, instead of away from it, and the film failed to answer one question sufficiently for me:  why didn't he just ditch the camera, which was slowing him down and took up one of his hands, and run?

I know I'm slating this film, but, apart from the criticisms I've levelled against it already, there are a few good points.  There's a section in the film where our characters are walking along a tunnel to ... well, I think they were looking for a safe way to get closer to the monster to look at it; I don't think they were trying to escape.  I must have missed the line that explained this entry in their list of poor decisions.  Anyway, they're traipsing along in the dark, when suddenly, a couple of mini-aliens (looking like a smaller, dressed-down version of the bugs in Starship Troopers) start attacking them, snapping and biting and scratching and generally causing lots of trouble.  One of the cast suffers a scratch or a bite to her shoulder, and, when the team take a proper look at it, it's infected and looking decidedly alien.  I surmised at this point that she was going to turn into an alien (something like District 9), but I was wrong.  The team make their way back to the surface, and are found by the army, with one soldier delivering the best line of the film, "We're not sure what it is, but we know one thing: it's winning."  

However, despite one of the team being seriously wounded, and is desperate for treatment, the characters decide they still want to go and rescue their friend (they assume that she's as daft as they are, and hasn't made a run for it).  Not, "Please will you treat our friend's shoulder," but, "We know we're unarmed, but want to go risk our necks too, where's the way out?"  Fortunately (for the story, not for the characters), one of the medical staff in the army's triage centre spot the wound, and with a shout of, "Bite!" they whisk the female lead (now bleeding from the eyes) out behind a screen, where she dies a very swift and dramatic death.  Still undeterred, our characters (I'm not calling them heroes, sorry) manage to get back outside, with the promise of a helicopter pick-up in the following morning.  They find their friend's apartment block, toppled over and leaning on another adjacent tower.  Has the friend run away?  Has she died?  Do we care?  Do the characters?  No.  They decide that it makes sense to go up the adjacent block and then jump across.  Oh dear.

At least the army and military have the right idea, as we see fewer armoured vehicles on the ground, and more aircraft firing missiles and dropping bombs, as our characters go on their crazy mission.  

I think I can summarise my disappointment or dislike of this film with one of the final scenes; the crew enter the tower block, and try the lifts.  They don't work, so they decide to take the stairs, and I suddenly realised the stupidity of what I could be about to witness:  a trio of people climbing all the stairs of a skyscraper.  Fortunately for me, Mr Cameraman pressed the pause button a few times, to save me from total boredom (but highlighting how bored I was at this point) and during one conversation on the stairs, he and his friend say, "What are you talking about?" "I'm just talking.  I don't know why I'm talking."  Sadly, sir, you're the only narrative to this story.  Otherwise, if you don't know why you're talking, then, even more sadly, neither do I.
Some of my other film reviews:
Cloverfield
Inception
The Green Hornet
Transformers 2: Revenge of the Fallen
Transformers 3: Dark of the Moon
Transformers: Bumblebee
Transformers: One
Tron

Web Analytics: Why are your pages getting no traffic?

I guess this comes as a follow-up to my first web analytics post - how to determine which pages on your site aren't getting any traffic.  The question I'm thinking about today is not which pages aren't getting traffic, but why not?  This comes after my own experience in discovering certain pages have no traffic.  That's not just low traffic, or less than I'd like, but none at all, which is very difficult to prove.  After all, our web analytics tools are designed to show some traffic, and finding the pages which aren't getting much is often a manual case of searching for a particular page's title, name, or URL.


But what if you can't find any traffic at all?  The request comes in from a stakeholder, "Please can you tell me how much traffic my page has had in the last month?"   Yes, it's a reporting question, and we'd like to analyse why the question's being asked, but let's fast-forward that part and just get to finding the traffic.  So, where is it?  There's none at all.  You can only show this because it's not showing up anywhere - Site Catalyst and Google Analytics won't report a zero figure against your page (unless you followed my suggestion a few months ago). 

This means you've reached a tricky point.  Do you really want to tell your stakeholder (who may be very senior) that their page has no traffic?  After all, some people regard traffic to their page as a number which is as much a status symbol as their salary, and so your news update isn't going to go well.  You may have to say it's had no traffic, but there are a few possible reasons why (some of them more palatable than others).


Possibility 1.  The page isn't tagged.


It happens.  Yes, we'd all like all our pages tagged, but occasionally one slips through the net.  Perhaps it's not a typical page built from the content management system's templates.  Maybe the tag got mangled by some other javascript on the page, or an agency or marketing tag, and it hasn't been firing properly.  This means no data - which is a different issue than no traffic.  Any solutions to this one are going to require a confession, and then some creative accountancy.


Fix 1:  look at other pages at a similar depth in the site (which are tagged) and see how much traffic they're getting.  If the page is accessed through the navigation of a page at a higher level in the site, look at the other pages in the navigation.  For example; if the navigation on your Widgets page reads:  Large widgets; Medium Widgets; Small Widgets; Micro Widgets; and you're missing data for one of them, then obtain traffic for the other three.  It would be convenient if they were in a trend, but that's not essential.  Let's say the 
figures are:

Large widgets:  800 visits
Medium widgets:  690 visits
Small widgets - data not available
Micro widgets - 400 visits

You can say that there were approximately 450-600 visits to the Small Widgets page during the period in question.  And no, it's not very accurate, but at least it's a start - it gives you an order of magnitude to work with (which may be enough for your stakeholder until you get the page tagged, double-quick!). 


Fix 2:  did the page have a tag prior to some sort of mangling?  Use a variation of Option 1 and follow the performance of the other pages since the mangling incident, then build up a trend during the 'empty' period.  Either way, fix the tag as soon as possible!


In either case, getting some traffic from now onwards should enable you to better model what the traffic was before it was tagged.  This assumes that you haven't done something like point a large amount of online marketing at it, which has now come to an end.  In this case, you can hope to use the marketing stats (clicks will do if all else fails) to build up some sort of picture.

Possibility 2.  The page actually hasn't had any traffic.

If not being tagged was bad, then this is potentially worse for your stakeholder - as we said, he's as interested in traffic to this particular page as he is in the performance of his stocks and shares.  Again, there are various reasons why this might be the case (and it's worth testing it from your test kit to make absolutely sure it's not a tag issue).

Possibility 3:  Is it actually possible to navigate to the page that's being investigated?  

It's one thing for a stakeholder to send you an e-mail asking for traffic figures for "http://www.mysite.com/my-precious-page.html" and yes, the page is there on the site, but how do you reach it from the home page?  Or from another landing page?  Has the navigation from the hub pages on the site been 'fixed' or 'improved' since the page went live, and has the page lost its source of traffic?  Remember that search engines spider your site through your links, so if the page has lost its links, and it's not got external links (from online marketing of any type), then you're like to lose traffic completely. 

Try navigating to the page, starting from the home page.  Then try navigating from the page in question - follow its navigation links, and in particular, its navigation breadcrumbs (if it has them) and work up and down the site hierarchy.  As you navigate up, towards the home page, are you still able to retrace your steps using the on-page links (without the back button)?  Time to check the navigation and make sure everything's right.


Just because a page is on your site doesn't mean it's getting traffic.  Sorry.


Possibility 4:  It's time to put on a crash helmet:  perhaps the page isn't very interesting or appealing. 

If you're looking for a get-out-of-jail card, it may be better to say that maybe the page isn't promoted well within the site, as the link text from other pages isn't very descriptive.  A page may very well offer interesting facts, opinions, views, get-quick-rich schemes and whatever else, but if the link text on the other pages in the site say, "Click here for more information" or "Find out more about this" then the wonder page is not going to get much traffic.  It depends on where the page is, what its content is, and, equally importantly - how easy it is to find on the site, and how well it's promoted.

I dare not say that an untagged page issue may well be masking problems with the site navigation or the page's attractiveness - just tackle the issues one at a time!

Other articles I've written on Website Analytics that you may find relevant:

Analysis is Easy, Interpretation Less So
Telling a Story with Web Analytics Data
What is "Direct Traffic"?
Reporting, Analysing, Testing and Forecasting
Pages with Zero Traffic

Film Review: Inception

A few months ago, I made a joke on Facebook that I was thinking about watching Inception but that the start didn't look very good (get it?).  However, Mr Lovefilm chose to send us Inception as our latest rental, and I sat down to watch it, although I couldn't recall when I actually added it to my selection.

I wasn't entirely sure what to expect from this film, but I am pleased to say I was very pleasantly surprised by it, from its high-paced start, through its plot development, outline of the 'rules' of the story through to the execution of the final mission.  The plot, in a nutshell, is that our main character, Cobb (played by Leonardo DiCaprio) must recruit a team and work with them to plant an idea in the subconscious of a highly powerful businessman (Cillian Murphy, from Batman Begins) while he's asleep.  There's some clever exposition which is subtle but clear, about how the team can break into a person's subconscious and steer his thoughts through a dream.  There are some extremely impressive visual effects - the sight of the whole landscape being folded over in a dream, and watching the characters walk across a horizontal and then vertical surface was especially eye-catching.


I also enjoyed some of the later fight scenes; in particular, those that Arthur (played by Joseph Gordon-Levitt), Cobb's right-hand-man had in the hotel sequence.  As the story develops, his world becomes one with zero gravity, and he has to fight one of the story's security guards in zero g.  The plot works well on a number of levels, cleverly identifying some of the strange effects we experience in our dreams - the sensation of falling, which wakes us up; the uncertainty of being in a dream or not; the difference in how quickly time passes in a dream, and being in a dream within a dream (waking up but still being in a dream).  It is these last two effects which provides the main structure for the story, as Cobb and his team sedate their victim and then get him to experience a dream within a dream, and then, within a dream again.  The story, script and visuals all help to make this complicated concept hold together extremely well, and even as the story becomes more complex (and the speed of the passage of time changes from level to level), I had no problem in keeping up with exactly what was going on.  The locations and scenes were all unmistakably different from each other, as the locations were lit and filmed in clear, distinct ways, each with their own sub-storylines.  The director deserves special credit for shooting the story in this way - it was consistent, logical and sensible.


The main plot - capture the businessman, infiltrate his dreams and plant an idea - is accompanied by a second plot, which is one of Cobb's own subconscious, as we discover that his wife has died, but that she's very much a part of his subconscious.  Through his guilt over her death, and her recurring presence in her dreams she becomes a real threat to their virtual plan. Cobb is called on this by Ariadne (Ellen Page, AKA Juno), the member of his team recruited to construct the dream landscapes that they use.  As the story develops, and Ariadne uncovers more and more of the truth behind the death of Cobb's wife, and how his memory's of her are affecting his subconscious, and, in turn, jeopardising the safety of the team.  The sight of a freight train ploughing through the middle of a busy city street is memorable, especially when it transpires that this isn't part of the 'victim's' mental defences.

The film, for me, was thoroughly enjoyable and highly engrossing.  There are strong Mission Impossible overtones throughout, and especially in the first half, as an action film.  It's got a strong psychological story (touching on the peculiar nature of dreams), and a character story (the development of Cobb and his layered character, and his inner troubles is well done without drowning the main story).  All-in-all, I would strongly recommend this film, although it will require (and reward) close concentration to fully grasp the story - I should know, I had to watch the first 15 minutes three times until I could see it uninterrupted, and it was worth it.

Some of my other film reviews:
Cloverfield
Inception
The Green Hornet
Transformers 2: Revenge of the Fallen
Transformers 3: Dark of the Moon
Transformers: Bumblebee
Transformers: One
Tron
Wing Commander
Pixels
Wreck it Ralph