Double-Digit Numerics

 

Is it likely that the All Blacks Win the Rugby World Cup? - No

The All Blacks may be the best team at the 2011 Rugby World Cup but it is still more likely that they lose the cup than win it.

Pardon?

This seeming paradox is all to do with the probabilities

(Paul the Octopus story moved to here)

 

What is the Probability that the All Blacks Win the Rugby World Cup - Simple Answer:

Let's do some simple back-of-the-envelope calculations.

Let's assume that the All Blacks make it to the quarter finals. It seems virtually certain that New Zealand and France will top Pool A and advance to the quarter finals.

For the All Blacks to win the cup they must win the next three matches in a row. What are the chances? Using the results of previous games to estimate probabilities we get these figures:

The quarter final match is very likely to be against Argentina or Scotland. Doesn't matter which because the probability that New Zealand wins is 0.96 in either case.

The semi final match is likely to be against South Africa and the probability of a win is 0.58.

The final match is likely to be against Australia and the probability of a win is 0.70.

The probability that the All Blacks win the Rugby World Cup is the product of these three probabilities i.e. 0.96 * 0.58 * 0.70 = 0.39

0.39 is a little better than 1 in 3 and is quite low. Hence our apparently paradoxical conclusion:

IT IS FAR MORE LIKELY THAT THE ALL BLACKS LOSE THE 2011 RUGBY WORLD CUP THAN THAT THEY WIN EVEN THOUGH THEY ARE THE BEST TEAM IN THE TOURNAMENT.

Using these numbers we would expect the All Blacks to win about a third of all the Rugby World Cups in which they are the best team. Since there have been 6 Rugby World Cups and the All Blacks have won one of them our probabilities may be a bit high. For example, in all RWC games the All Blacks have a 83% success rate. In all games against the top teams they only have a 50% success rate (just 12 games though).

Sunday Star Times article that refers to a previous calculation of these odds

 

What is the Probability that the All Blacks Win the Rugby World Cup - Better Answer:

The probabilities above are a bit rough and we can do much better. For example, if England beat Australia then New Zealand will meet England in the final and will have a higher chance of winning. So we might have estimated a bit low by using Australia in the final.

Here are some better probabilities of each team winning the Cup:

With the probability that the All Blacks win being only 0.472 it is more likely that they lose than win.

(Note that the probabilities sum to more than 1. That's because only one of Argentina or Scotland should be in there. We'll remove the appropriate team when the pool results are known)

18 September update: Ireland beat Australia. So it looks like Ireland will finish top of their pool.

2 October update: The pool play is over. We now drop Scotland out of our calculations. Our final knockout stage probabilities are:

9 October update: After the quarter-finals

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Who will be in the finals?

As of 2 October: The probabilities of the various match-ups are below. Note that these probabilities are estimates except where the probability is zero which is an exact probability. This is due to some finals combinations not being possible. Don't bet on a New Zealand-Australia or New Zealand-South Africa final because it can't happen.

As of 9 October:

16 October update: Not much to say now. This is our last update. The probability that New Zealand beats France is, under our model, 0.73. Not as high as many would suppose. The TAB is paying $7.00 for a France win. Might be a good bet eh?

 

How We Calculated the Probabilities

This gets a bit technical from here.

We need to take into account every possible game combination. To do this we need to make some more assumptions about who gets past the pool stages. Once we know who actually does we will update this page and put in the exact numbers.

Suppose New Zealand, France, England, Argentina, Australia, Ireland, South Africa, and Wales make it through and finish in that respective order in the pools. (If we replace Argentina with Scotland then the probabilities don't change).

What does change the numbers a bit is if South Africa finish second in their pool. If they finish first then they cannot play New Zealand in the final (since they meet in the semi final). If they finish second then New Zealand gets an easier route to the finals and has a bigger chance of winning. But we'll cross that bridge if we ever get to it.

We use the results of 4054 test matches between these 9 countries to estimate the probability of each team beating the other. We apply a James-Stein estimator to these probabilities to give even better probabilities. (The reason for the James-Stein estimator is too complicated to explain here). We get the following table. This gives the probability that the team on the left beats the team on the top.

Then we went through every possible game result and calculated all possible 128 results and their probabilities. That gave us the final probabilities above.

What about the home game advantage? Should we increase the probability of New Zealand winning because they are playing at home? Probably. New Zealand has played better at home than away. But then you could argue that we should decrease the probability of New Zealand winning because they usually "choke" at the Rubgy World Cup.

It is difficult to assess these psychological changes in probabilities so we remain more objective by ignoring them.