The Chancellor, the Right Honourable Philip Hammond MP, recently stated that he would not be providing funds to put in place contingency measures, to prepare for the outcome of the Brexit negotiations being "No Deal".He did not want to spend money that could otherwise be spent on hospitals, schools, defence etc on protection against a merely hypothetical outcome.

Was this wise?Or was it naïve optimism?Of course, no one can say with absolutely certainty.But we don't have to rely on gut feeling here – mathematics can help to provide some bounds on the probability of a Deal or No Deal outcome.

Cast your mind back to your schooldays, when you were asked to calculate the odds of a series of coin tosses all being heads.

For the first toss it is simply ½ = 0.5 or 50%, since there is an equal chance of the coin landing heads or tails.

For two tosses in a row returning heads, the probability is ½ x ½ = ¼ = 0.25 or 25%.

For threes tosses in a row, the probability is ½ x ½ x ½= 1/8= 0.125 or 12.5%.

The rule is that the probability of a series of events is simply the product of the probability of each individual event.In this case, the probability is just ½ x ½ x ½ … as many throws as you want.

For any Brexit Deal (whatever the detail of the Deal) to be approved, each and every one of the 27 remaining EU countries has to vote 'Yes'.Armed with our knowledge of how to calculate the result of a series of individual votes ('Yes' or 'No') we can calculate the probability of all 27 countries in the EU voting to approve a Deal. To keep things simple, let's assume that each and every country has the same probability of voting 'Yes' to a Deal.

Let's also assume that each country is genuinely free to make its own decision; that no single country (for example, to choose a country at random, Germany) can impose its will on the other countries.And that, despite advice from some authorities or experts (for example the European Commission), each country will in the end decide for itself how to vote (Yes or No to a Deal).

We just multiply the probability of each country voting 'Yes' 27 times, once for each country that is voting.We can, of course, choose / guess what we think is the probability of each country voting 'Yes', and then look at the results for different values of this probability.

When we do this, the results are quite sobering – and they should give The Chancellor, the Right Honourable Philip Hammond MP, pause for thought.

Even if we assume that each and every one of our EU friends is 99% likely to vote Yes for the Deal (whatever the details of that Deal), the combined result is … only 76% probability of Yes.If the probability that each and every one of our EU friends is 98% likely to vote Yes, the combined result is 58%.

The table below summarises the results for a range of probabilities.

Oh dear!Even on some pretty benign assumptions – for example that our friends in the EU are very likely to vote Yes to a Deal – the results are worryingly unlikely.

How often, in your experience, has a politician or an economic forecaster or a respected broadcaster been able to assess the outcome of an event (an Election, or a Referendum, or the unemployment rate, for example) with 90% accuracy?Yet even if Mr Hammond is correct in assessing the probability of each of our friends in Europe voting Yes for a Deal as 90%, there is only a roughly 1 in 20 chance that the end result will be Yes to a Deal.

Of course, this assessment can be criticised as being simplistic.Which, of course, it is.

Because in addition to each of the 27 countries voting Yes, any Deal needs to be approved by our own House of Commons.There are many MPs who might be tempted to vote No, for any number of reasons.For example - because they actual want a No Deal – or because they want to 'Exit Brexit', in defiance of the result of the largest exercise of democracy in the UK's history.So there is no certainty that our House of Commons would vote Yes.

And, of course … it may be that individual regions within the EU countries would vote against the Deal.Remember the EU trade deal with Canada, many painstaking years in the making, that was delayed because the Belgian region of Wallonia vetoed it?

But let's not be too gloomy – I am not sure that the regions will have a vote.But then again – the European Parliament will.

Oh dear …

For those who want to see the detailed workings, I have embedded an Excel spreadsheet in this document:

xlsx
File Name: Worksheet-in-How-Likely-is-a-No-Deal-for-Brexit.xlsx
File Size: 15 kb
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By: Paul Hicks