Rabbit’s Card Puzzle

Two people sit down at a table, each with a shuffled pack of standard playing cards in front of them.
They each turn over the top card and place it face up on the table. If they have the same card they shout “SNAP!”

For this to occur both number and suit must match. Four of Spades does NOT match with Four of Hearts.

The chances of this happening on the first go is 1 in 52.

Now more people want to join this fantastic game. Each time a new player joins they bring their own pack of shuffled cards.
i.e. When there are 4 players, 4 cards are turned over. If there is a pair anywhere among the 4 cards they all shout “SNAP!”

As the size of the group increases, the chances of shouting “SNAP!” increases.

At some point the group becomes so large that it is more likely than not that “SNAP!” will be shouted when they each turn their first card over.

What is the smallest number of players required to make it more likely than not that “SNAP!” will be shouted when they each turn their first card over?

I will donate £10 to the chosen charity of the person with the first correct answer.
They need a rough reason though. You can’t just guess 1,2,3,4 etc. until you get the right number.

Please put your answer in the comments on this post, not on Twitter.

Good luck!

RedEaredRabbit

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And Now A Small Confession…

Last night, I published the results of my eagerly awaited jazz-sushi survey, where I attempted to find whether or not there was a correlation between liking jazz and liking sushi.

If you have not already done so you can read it here. (Feel free to skip the numbers bit if that stuff bores you.)

So I proved that a correlation existed and made it into a law and all is well. Well not quite. I should come clean about something. It didn’t really prove anything and I’ll explain why.

Firstly, (as many people pointed out), my questions asked for Yes/No answers to complex questions. There are lots of different types of jazz and usually someone doesn’t like or dislike all of them. My survey forced them to interpret the question as they saw fit. Worse, it caused people to give me long-winded answers which I had to interpret.

Why’s that worse? Well I knew what I wanted the outcome of my survey to be and while I didn’t consciously seek to influence the results in this manner, I am hardly in the best position to be a neutral judge.

Also as @mapsadaisical quickly pointed out, I had a self-selecting sample. This means people were free to choose whether to take part or not. Why is that bad? Well people knew that I was trying to find a correlation between people who liked jazz and people who liked sushi. When people know what is trying to be proven it influences whether they respond or not.

On Friday night I did my sums and found that there was a correlation but it was not significant enough to prove anything one way or the other. I explained this on Twitter and asked for some more responses. Of the next 12 responses 11 were either likes both or likes neither. This wasn’t coincidence, it was simply people wanting to help me show a correlation. Those who did like both or neither kindly though “I’ll help you out.”

Another example of this came when I was watching a morning day time TV show a few years ago. It was GMTV, or Anne & Nick or Richard & Judy or some bollocks, and they had a phone in poll. A phone in poll is even worse for this problem than Twitter because the effort of making the call is greater and they charge you money for doing so. You aren’t going to bother voting unless you have some compelling reason to do so.

The poll asked people to vote on whether or not they were currently in an abusive relationship. About 50% said yes. At no point did the programme mention that the surprisingly high result could be influenced by the fact that this poll was much more important to someone in an abusive relationship and they were therefore more compelled to vote than someone who wasn’t. In fairness to the programme they didn’t try to conclude that 50% of all relationships were abusive.

There is another problem with the way in which I gathered the stats. Even if everyone who saw the question had responded, I didn’t survey a proper cross-section of the public. Supposing I did a poll on Twitter to find out whether people thought Social Networking sites were a good thing. I would certainly get a higher proportion saying Yes than if I stopped people in the street and asked. Although there is no obvious reason for people who use Twitter to have different views on jazz/sushi to the public at large, the whole experiment was to find a correlation between two seemingly unrelated things so really I should have excluded any other similarities between the respondents.

A good example of this is in the polls which newspapers do on their online websites. If the Daily Mail asks a question about immigration on its website is the response going to reflect the views of the country at large? Probably not, because people who read the Daily Mail website are likely to have different views on immigration than the average person on the street.

You should treat with skepticism any survey that can’t show clearly how it gathered and interpreted its data to avoid external factors like this affecting the results. Companies like Ipsos MORI go to huge lengths to try to minimise these problems. I didn’t and as such you should just interpret my survey as a bit of fun.

RedEaredRabbit

The Jazz Sushi Survey

When I was just a young rabbit, I was taken, as a treat, to see the National Youth Jazz Orchestra who happened to be playing in my village. It was an epiphany and I was transfixed. Never in my life had I imagined music could be made so utterly awful. Equally shocking was that a good many people around me seemed to be enjoying it, and not just a little bit either. A ginger man a couple of seats away with his eyes closed looked for all the world like he was having an orgasm for the entire concert and for all I know he was.

Years later, I was having a pint with a mate in our local pub, The King of Toss, near Marble Arch. Above the King of Toss was a restaurant to which neither of us had paid any attention in the years since we’d been drinking there. Seemingly no one else had been paying it any attention because on that night a member of the waiting staff came into the pub with a plate of sushi, offering free samples in a bid to entice some drinkers upstairs. So I tried some. This, ladies and gentlemen, was my second epiphany. Never in my life had I imagined food could be made so utterly awful. How could it possibly have been that bad? After all, I like rice, I like fish. In fact given the same ingredients I could probably have made something quite nice. This was anything but. The rice has a texture like it had been cooked the day before, left in the pan to dry then scraped off. It was topped with little red things which seemed to have been sprayed with essence of unwashed genitalia and it was wrapped in one of those unbreakable plastic ribbons that bind up telephone directories. Bizarrely my mate liked it.

At some point in the years since, it occurred to me that I thought about jazz and sushi in pretty much the same way. Not simply in my dislike for them but in the way that I just didn’t get them. I knew plenty of people who were enjoying these pleasures and I would never be able to understand why.
I don’t like Crufts but I can understand why people like it. They get to see the most classically beautiful dogs all standing in a row. I just prefer dogs when they’re fetching sticks and eating slippers but that’s just my preference and I understand theirs.

Jazz and sushi were incomprehensible to me though and the more I thought about it, the more I wondered if these two seemingly unconnected things were in fact connected through people’s preferences. i.e. was there a correlation between people who liked jazz and people who liked sushi? Were these two things completely unrelated or was there a disproportionately high proportion of us who liked both or disliked both compared with the proportion of people who liked one or the other?

This previously unidentified correlation has been an untested theory of mine in the years since but then came Twitter and suddenly I had the perfect opportunity to test it out.

Last week I asked people two Yes/No questions:

  • Do you like jazz?
  • Do you like sushi?

And thanks to those who responded and retweeted it I ended up with 112 responses.

And so to the numbers. Firstly, I worked out the proportion of people who like jazz and the proportion of people who like sushi. The results were:


Using these numbers, I worked out my ‘null hypothesis’. i.e. what the results would be if there is no correlation.
i.e. of the 112 respondents, if there is no correlation between liking jazz and liking sushi then the proportion of people who like sushi and like jazz is:

112 x (64.6% x 54.87%) = 40.41 people.

The full results of this are:

Then I compared this with what the 112 people actually said:

Interesting… there are more people in the like both and like neither than there should be if the null hypothesis is true. Sure enough when I calculated the correlation it came out at 0.17.

Correlation is expressed as a number between -1 and 1. A correlation of 1 means that the correlation is perfect i.e. for me to get a correlation of 1 everybody who liked sushi would have to like jazz and everybody who disliked sushi would have to dislike jazz. A correlation of -1 represents a perfect negative correlation. In my case this would have meant that everyone who liked jazz disliked sushi and everyone who disliked jazz liked sushi. A correlation of 0 would mean there was no correlation at all between the data. My correlation looked like this:

So I had a correlation and better still it was a positive one, but although my figures had a correlation could it just have been I got lucky?

To determine this I needed to work out what the probability of this happening by chance would be if the null hypothesis were true.
I decided to use a fairly standard way of testing significance – that the probability of such an outcome would have to be less than or equal to 1 in 20. i.e. if there is no correlation then results as convincing as mine could come up no more than 1 in every 20 repeats of such an experiment – a significance level of 0.05.

Therefore, if the probability of my set of results coming up is greater than 0.05 then the probability of it having happened by chance is too great, my correlation is not significant and my results are inconclusive. If the probability is less than 0.05 then the chances of this having happened by chance are negligible and my correlation is statistically significant.

Are you ready? Drum roll, please. The probability of a correlation as pronounced as mine having happened by chance is……..0.045!!

That’s right, I really did it. I really did find a correlation between liking jazz and liking sushi. The theory I have held for ages has at last been proven.

I am not going to call it RedEaredRabbit’s Law. After all it is too important to be just mine – it should belong to all of us. I am instead going to call it Cole’s Law. (Nothing to do with Cole Porter  – I’ve just always wanted to call a law that.)

At some point I’ll explain why my method of gathering the data wasn’t perfect but for now I’m just going to bask in my glory.

RedEaredRabbit

P.S. I didn’t mean to imply that jazz and sushi were awful in absolute terms. Just that I dislike them and am personally unable to appreciate them. Don’t lynch me, please.

P.P.S. Please also read the follow up post to this survey here.

When the History Grad took on the IFS…..

It was interesting to read today’s publication from the Institute for Fiscal Studies regarding their analysis of George Osborne’s emergency budget. At the budget, you may remember, George Osborne presented his policies and stated that they would proportionally impact the poor less than the rich.

I like the IFS because it is independent of any political party but is extremely well equipped to analyse their economic policies and give us a viewpoint unbiased by any political persuasions.

The IFS have spent lots of time looking at George Osborne’s policies and doing their own sums. They have included lots of things that George Osborne didn’t include in his model. Things like the cost of mortgage payments do actually affect poor people, as do cuts in housing benefits and tax credits. They have also included the years 2013 and 2014 in their analysis which were missing from George Osborne’s.

At this point, I would have liked the government to thank the IFS for their analysis, review it in detail and decide, based on this review, whether or not they should change their policies. This wasn’t what happened. Within hours, the government had given a press release stating that the IFS had missed some important things from their analysis, such as economic growth and if they had included these they would have come to a different conclusion. This is a bit odd, because the IFS have included more things in their analysis than the government did in theirs. They haven’t as far as I can see missed out anything which they government included in their model, they have just added things the government forgot to include. To my mind, this doesn’t make it a worse analysis, it makes it a better analysis.

If George is going to get into a verbal ruck with the IFS about economics, I worry his modern history degree won’t help him out much but there is a bigger concern that I have. For the government to have so quickly found a the flaw in the data presented by the IFS they would have had to take the IFS model, incorporate the things they felt had been missed and then recalculate everything on that basis. i.e. they would have to have an even better analysis already prepared and ready to go. If they have this analysis then they should publish it so the IFS, you, me and everyone else can read it and respond. I suspect this analysis has not been done and their reaction is purely a defensive one.

In an ideal world a government would form policies by gathering the available evidence, analysing it and then determining the best policy based on that analysis. Before they implemented it they would determine the way in which they would measure its success or failure and if it were not behaving as expected they’d adjust the policy accordingly. This might seem an unattainable idealism but actually we are all doing such an exercise in our every day lives all of the time.

Supposing you live in Wimbledon and you get a new job in Canary Wharf. You now have to decide how you will get all the way across town and back every day. There is nothing direct so you have lots of combinations of options. You could take buses, tubes, mainline trains or even river boats.

You start off by typing your journey into your iPhone app. It suggests that the quickest route is taking the District Line through Earl’s Court, changing at Monument and taking the DLR. You try this for a while but realise that every morning you get stuck outside Earl’s Court for 20 minutes because the people who manage the arrivals and departures there are half-witted. The model on your iPhone app didn’t take this into account so in this situation you would try a couple of the other suggested journeys a few times and after a while, based on your experience, you’d settle on the route which worked the best for you.

The government equivalent is to decide that on the first day of work, they need an emergency iPhone app journey plan. After this, no matter how inconvenient it becomes they will stick to the route they took on the first day and claim it is the best.
When newer better apps become available they accuse them of not including something their original model had never included anyway like “leaves on the line” and stick to sitting outside Earl’s Court for 20 minutes every morning, pretending they meant to do it.

No one would go in for this nonsense with their journey to work so why do politicians insist on it for something as important as the economy? The difference is this:

How much would it cost you to admit you were wrong?

Sadly we live with a political system which overly punishes this natural human trait. When the apple fell on Newton’s head his reasoning didn’t go like this:

I need an emergency gravity law. It is very important to get this out asap. Thinking it through would be an unnecessary waste of time.

…and then…

My emergency gravity law is that apples are attracted to heads, through a strange new force.

And then refuse to change his law when someone pointed out it was a bit more generic than that and in fact everything was attracted to everything else.

Of course he didn’t. Science would hardly be where it is if all scientists were to all insist that the first thing they ever thought of were the absolute truth. Politics a bit different though. For one there is the opposition. Imagine there were someone employed for the sole purpose of taking your job off you. If there were, would you want to admit you’d got something wrong? Also there is the press. Every newspaper has a political agenda, and while I think being able to learn from experience and adjust accordingly is a good trait, the press seem to call it a “U-Turn” and think it makes you a weak politician, not fit to do the job. All this gives an incentive for governing politicians to refuse to admit their mistakes and pretend the policy they first thought of was the best one.

I admit though, the comparison with Newton was harsh because politicians are under a lot more time pressure than Newton was. As important as Newton’s theory of gravity turned out to be, it wasn’t as though anyone at the time was having a terrible time directly because they didn’t have an equation to explain why they were sticking to the Earth. The new government didn’t have that luxury. When they came to power they were under immense pressure to put in place some policies to start addressing their finances – after all they had promised to do so and been elected on that basis. I don’t have a problem with them doing this, as long as they could have a process to continually review what they were doing, take on board other people’s opinions where necessary and adjust their strategy accordingly when they were wrong.

The reality is a long way from this though and the best model available to us today, from the IFS, suggests that our current economic policies are punishing the poor more than the rich. The government deny this but haven’t produced a better model to show how they can be so sure.

Would it be so bad to have a political system where a government could take constructive criticism of their policies into account and improve upon them because of it?

Wouldn’t that benefit to the country as a whole?

Would it really be so terrible of them to at least read the IFS publication with an open mind before responding?

I don’t think it would be terrible. In fact, I think they should read it. It is rather good.

RedEaredRabbit

Blame it on the Bonus?

It seems to me that politicians are very keen on blaming the recent financial crisis on the bankers who earn big bonuses.

I rather think it is a little more complicated than that but before I stray too far into why, I’ll give a basic example of why a trader may tempted to take a risk.

I recently found out that Scottish Power have overcharged me on my direct debit by so much for so long that they owe me £1,000 and have to send me a cheque. I could invest this in a savings account with a high street bank. I may get an interest rate of 1% on such a deposit, meaning in 1 year’s time I will have £1,010. The £10 I have made doesn’t really set my world on fire (especially when taking inflation into consideration I will have made a loss) but the upside is my money is safe. It is so safe that it is even guaranteed by the UK government in the event of the bank going bust.

Alternatively I could invest my £1,000 in the stock market. The stock market is much less predictable – my money in a year’s time could easily be £1,250. It could easily be £800. If things went really badly for the company I invested in it could be worth £0. In fact I have very little idea about how much it is going to be worth but returns in the stock market historically outperform returns on a bank account so I may be tempted by the risk.

This is also the reason why a trader takes risks in a bank. Simply, risky investments are more likely to yield a larger return. If there were a risky investment which had a likely lower return than a safe investment no one would bother going near it. Therefore we can say that when a trader takes a risk they think it is more likely to yield a larger reward than the safer option.

Now let’s extend this principle to Evelyn. Evelyn is an evil, heartless trader who, when she isn’t out running over old ladies in her Ferrari, has a bonus scheme which pays her according to the profit she makes for the bank. If she put all of her available funds into a safe bank account she’s going to get no bonus – anyone could have done that. In order for her to get the new Lamborghini she’s got her eye on she is going to have to take some risks.

Evelyn has taken risks with the money for the last few years and every year the risks have worked out and she has made a fortune for the bank and a fortune for herself. Until 2008. In 2008 everything didn’t work out and she lost 100 fortunes for the bank. The bank couldn’t foot the bill and the tax payer had to bail it out. Therefore Evelyn caused the financial crisis.

It was all Evelyn! Case closed, right? Wrong. Who spotted the real problem in the above paragraph?

“…and the tax payer had to bail it out.”

It may not seem immediately obvious but Evelyn hasn’t actually done anything wrong in all of this. All she has done is respond to her incentives. She knows the riskier the strategy the more chance she has of making the big bucks. The smallest her bonus can be is zero – if her strategy doesn’t come off it’s not like she has to fund the loss herself. She has simply responded to the incentives the bank gave her.

In a bank as well as the traders, they have people called risk managers. Risk managers are responsible for determining what traders are allowed to invest in and how much they are allowed to invest in it. They do lots of complicated maths and put in place policies to police the traders. If I were going to start pointing fingers at bank employees I would probably have a good look at them before the traders. I’m not though.

Recently @WH1SKS, (one of the greatest people on Twitter, follow him) said he thought that the banks didn’t seem to have really paid for their failure, although everyone else did seem to be paying for it. He was completely right.

Banks, you see, are “too big to fail.”

“Too big to fail.” It drives me nuts. Outside the financial sector you will find no one “too big to fail” and you will find no one who could possibly fail in such a big way as they have.

I work in a small company. If we made enough bad decisions we could probably make ourselves go bust. At that point we could go to the chancellor and ask for a bailout but we won’t get one because outside our staff and our clients no one gives a stuff whether we’re there or not. Our company therefore has a massive incentive not to take unnecessary risks – a bunch of risky strategies could be the end of us. A risky strategy for a bank now means either a massive profit or, if it all goes tits up, a handout to keep it going. The bank is now no different to Evelyn – in the good years make a bundle and in the bad ones know your maximum downside is you keep going anyway and someone else pays for it.

The banks could not be allowed to go bust because the impact on the global economy would have been far worse than it was to bail them out. They each had so much in the way of liabilities that them going bust would not only have taken out the finances of many individuals and companies, it would also have taken out other banks and the whole thing would have gone down like dominoes.

So what’s changed? If RBS tomorrow were to announce they were in a pickle they’d get bailed out again because the same problem is true today. If the banks were too big to fail before, it’s even worse now because due to the mergers which followed the financial crisis, they are bigger now than they were before.

All this has proved is that it is a completely unworkable system to have organisations which cannot be allowed to go bust when they make bad enough decisions. If that is the case, they have no incentive to abandon risky strategies and they will continually need to be bailed out when the strategies don’t come off.

So what’s the solution? There are several contenders. Perhaps banks should have to provide the impact of their potential bankruptcy as part of their financial reporting and auditors should have to verify they could go bust without causing financial meltdown and if they can’t prove it they would be broken up. Perhaps they just need to hold more capital? Perhaps there should be legislation forcing them to raise more money through equity rather than debt?

It’s a debate that needs to happen because it is a problem that must be solved and has not been solved. By bailing them out all we have done is put an Elastoplast over the underlying problem. There are still financial behemoths out there with incentive to take risks and nothing to guarantee it won’t result in a bailout. I don’t know the full solution, but I do know one thing:

If a bank is too big to fail – it’s too big.

RedEaredRabbit