Rod Crosby asks: “Can we measure the probability of such an outcome?”

Electoral System: The first graph has been compiled from Anthony Wells’ data for the new constituency boundaries. For each percentage point the LibDems might reasonably score in the range 15-25%, the outcome of the Election has been calculated for plausible Conservative LEADS ranging from 0-11%. The concept of Swing implies that the difference between the main two parties is important while their absolute level of support is not. So we can represent three vote variables with only two dimensions.

LibDems Kingmakers: The yellow region in the middle of the graph forms the area where the Liberal Democrats could combine with either main party to form a MAJORITY, irrespective of which party has the most seats. The black line that roughly bisects this area – at around a 6% Tory lead – is the point where the Tories would become the largest single party.

Labour or Tory Minority: The pink and pale blue areas are the regions where the LibDems alone could not combine with the second party to form a majority, resulting possibly (but by no means certainly) in a minority government formed by the leading party.

What is immediately apparent from the graph is the huge area which comprises a Hung Parliament in some shape or form. i.e. ALL AREAS EXCEPT THE DARK BLUE AND DARK RED AREAS.

How big is this area? The perhaps surprising answer is fully 75% of the battleground…

In other words, if we consider the graph as a DARTBOARD, and the Electorate a not very accurate dart-thrower, the chance of the dart landing in the “Hung Parliament” region is about 75%. This remarkable electoral landscape is unprecedented, and is a startling feature of contemporary British politics – with profound implications.

Opinion Polls Say we have an opinion poll of a thousand people which gives the Tories 36%, Labour 34%, LibDem 19% and others 11%. What can we deduce from the poll?

Often, we talk about something called the Margin of Error. In this case the margin of error for the top two parties- at a 95% confidence level- is about 3%. This means, if 100 similar polls were taken we would expect that about 95 of them would estimate a party’s support to within +/-3% of its true value.

So, we are 95% confident that the actual level of support for the Tories lies in the range 33-39%, and Labour is in the range 31-37%. Sometimes, particularly in the US, such polls are termed a “STATISTICAL TIE”, because Labour “could” be ahead of the Tories by 37-33%. However, this interpretation can be misleading…..

A better question is:- What is the chance that the Tories are really ahead of Labour, be it by just 1 vote or 10 percentage points? Probability theory gives the answer as 78%.

We can extend these principles in two ways.

Firstly, by combining more than one poll we can reduce the Standard Error. For example, say we had three polls taken the same day giving Tory leads of 1%, 2% and 3%. The average lead is 2%, the same as our single-poll example above. However, we can now say the chance the Tories are ahead to some extent is about 90%, an improvement over the 78% calculated above.

Secondly, using probability theory we can also come up with the odds that the Tories are ahead of Labour by some value X – say by more than 6% or by less than 3%. In our three-poll illustration we can compute the chance of the Tories being ahead by more than 6% as very small – less than half of 1 percent. On the other hand, the chance of them being ahead by less than 3% (including a small chance of being behind) is quite high – about 74%. It follows that the probability of a Tory lead of between 3% and 6% is about 25%.

Putting it all together
So, we can throw some current opinion poll “darts” at our electoral system “dart-board”, and use probability theory to calculate the odds that the real position is inside the Hung-Parliament region.

As we can see from the first graph, the lead the Tories require to form a majority ranges from about 8% to 11%, depending on the LibDem level of support. Likewise, a small lead in the range 2.5% down to zero would result in a Labour majority.

In other words, we will compute the probability that the Tory lead is neither so high as to give a Tory majority, nor so low as to give a Labour majority, that is – THE PROBABILITY OF A HUNG PARLIAMENT. The results, based on 78 post-Cameron polls, are rather interesting….

Since May 2006, with one exception, a hung Parliament has been overwhelmingly likely, with probabilities in excess of 90%. The “exception”, on 16/10/2006 is quite possibly a “rogue” poll, since probability theory also tells us to expect 3 or 4 rogue polls in this dataset of 78 polls.

In a nutshell, the Tory lead has been consistently slap-bang in the middle of the wide hung parliament band, and statistics say it is extremely unlikely they are either far enough ahead to form a majority or so slightly ahead that Labour could form a majority.

We can also use this technique to calculate the odds that the Tories would be the largest single party, or that the LibDems might be Kingmakers. The current figures are 2% and 27% respectively. The chance of Labour being the largest party in a hung parliament is about 94%, although the probability the Tories are ahead in votes is very close to 100%.

Rod Crosby is a regular contributor to PBC discussions

Betting note by Mike Smithson. The main market where you can bet on whether the next election will produce a Labour majority, a Tory majority or a hung parliament is at Betfair. The current price on the hung parliament option is 1.38/1 – so if Rod’s approach is right this makes a great value bet.