Win the toss win the match?

In watching this world cup, there have been 6 sides who have looked a step ahead of the others: Australia, England, India, New Zealand, Pakistan and South Africa. There have been 8 matches featuring two of those 6 sides.

Pakistan beat India. Pakistan beat New Zealand. New Zealand beat India. England beat Australia, Australia beat South Africa, South Africa beat England, New Zealand beat England and Australia beat Pakistan.

Interestingly, in 7 out of those 8 matches, the team who won the toss, won the match.

If the toss was independent, the chances of this happening are about 3.5% (9/256 for anyone who wants a more precise answer). That is very unlikely. But it is not so unlikely that it would be considered impossible.

If I throw 8 coins, every now and again it will come up with either 7 heads or 7 tails.

It made me wonder if the toss was a significant contributor to team's success, and if so, what can be done about it.

The first thing to do is to build a model to predict the outcome of matches. This is important, because England beating Papua New Guinea after winning the toss does not say much about the importance of the toss, because England would probably beat Papua New Guinea if they lost the toss too.

I decided to use logistic regression to build the model. I looked at every international match between 2 sets of competent (or semi-competent) side in the last 3 years. Looking back afterwards, I noticed that I had left out Nepal (who did deserve inclusion) but they were the only team that the ICC currently have ranked in the top 20 that I left out. I also included a few lower ranked sides in order to give a better picture of the difference between the teams near the bottom of the rankings for the T20 World Cup. So I also included the likes of Singapore, Kenya, USA and Malaysia.

I chose to use logistic regression because it has been helpful in the past for giving realistic probabilities of winning for limited overs matches. When I tested the model, it explained about 85% of the variation in results. When it said that a team had a 50% chance of winning, they generally won about 50% of the time. When it said that a team had a 70% chance of winning, they generally won about 70% of the time.

It is not perfect, but it is simple enough and close enough to tell us about the impact of the toss.

The factors that I used were the team, the opposition, and if the match was home, away or neutral.

The model suggested that the 6 teams that I listed above, along with Afghanistan were the 7 best teams. It probably overstated the strength of Afghanistan, due to them not playing at home at all, and so therefore missing out on the home advantage. Their players are so familiar with their adopted grounds that they have an advantage there that is not accurately reflected in the tag "neutral."

Once I had built the model, I could then make predictions about all the world cup matches.

In this graph, I have the modeled probability of winning on the x-axis and then the actual outcome on the y-axis. The green points are where a team has won the toss and the red ones are where they have lost the toss.

I have divided the data into 3 groups - expected loss, too close to say, and expected win. The numbers are the proportion of wins by the team that won the toss (green) or lost the toss (red).

We can see that the team that won the toss has won more than the team that lost the toss in each of the 3 regions.

This is fairly compelling that there is an advantage in winning the toss. But it is not nearly as dramatic as 7 out of 8 in the first sub-group that I looked at.

This made me wonder if there was some sort of accidental gerrymandering with the way that I selected the data. So I tried 5 groups instead of three.

This time I grouped them together, and looked at the expected number of wins against the actual number of wins.

This time I added in two parallel trend lines, and looked at the difference between them. The groups of teams who won the toss ended up winning about 1.5 more matches than the groups of teams who lost the toss.

This was interesting, but I was not sure what to read into this. So I decided to re-randomise the toss, to see what would have happened with an independent toss.

To do this I randomly assigned to each match one team as the designated toss winner. Then I redid the groups, and saw what the difference was. I wanted to know how rare a difference of 1.5 was.

It turned out to be more common than I would have expected. After 10000 trials, I found that roughly 10 % of the sets had a difference of more than 1.478, and roughly 10% had a difference of less than -1.478. For a re-randomisation, 10% is about the cut off where you say that it was likely or unlikely to have been caused by natural variation. 

This was a surprising result. I was expecting to find that there was clear evidence that winning the toss improved the chance of winning, but instead I found out that it might do, or it might be just natural variation.

There are two major errors in statistics: saying too much or saying too little. This situation looked like one that had the potential to put egg on my face no matter which way I went. There was not quite enough evidence to be very confident that the toss made a difference, but there was also enough evidence to be quite confident of that fact.

I wanted to try one more test before I decided that I didn't know what to say.

This time I picked 60 random innings from any match in the past 3 years. I applied the model to that innings, and then grouped the innings and found the difference between the two lines. I repeated that 1000 times.

This time I found more like what I was expecting.

Less than 1% of the randomly selected innings had the impact of the winning the toss as high as it has been in the world cup. Interestingly, the teams that lost the toss actually had a slight advantage (1.7% of the time losing the toss had an advantage of 1.478 or more matches)

This tells us two interesting things:

Winning the toss seems to have given teams an advantage in this world cup, and it does not normally give teams any advantage whatsoever.

I wondered if that was due to the dew factor. It can often get harder to bowl as the match wears on due to dew in the gulf states. But that does not seem to have been the difference. Winning the toss was roughly as much of an advantage in the daytime as it was in the nighttime.

The biggest single factor seemed to be Dubai International Stadium. Matches there seemed to be much more toss dependent than almost anywhere else.

And that is where the final is being held.

Given that, the toss is likely to give an advantage to whoever wins it. Or perhaps it will revert to type, and there will be no advantage.

Assuming that the toss should be factored in, my model has the following probabilities for the final:

If Australia win the toss: Australia 67%, NZ 33%.

If New Zealand win the toss: Australia 30%, NZ 70%

Neither team is 100% or 0% in either scenario, but there's clearly an advantage.

Now it will be up to the players to see if they can overcome it.

Some Questions ahead of the 3rd test

Questions leading into the third test

1. Who will actually be fit to play for New Zealand?

There is talk that Kane Williamson, Henry Nicholls and Mitchell Santner were all too sick to get out of bed yesterday, and all are unlikely to play. Trent Boult and Lockie Ferguson have already gone home. Glenn Phillips has been called in as a late replacement, meaning that there is a chance that New Zealand will end up playing four wicket keepers, and recalling Jeet Raval to the squad simply due to lack of other options. If those 3 are all out of contention, then New Zealand’s top 7 is likely to include Raval, Tom Latham, Tom Blundell, Ross Taylor, Phillips, BJ Watling and Colin de Grandhomme.

2. Will either side opt for two spinners, and if so, who will make way?

The Sydney Cricket Ground has a reputation as a spinners track, and both teams have added an extra spinner into their squad. If Australia opt for Mitchell Swepson, then they are likely to end up either dropping a batsman, or going in with only two pace bowlers and giving the 3rd seamer role to Matthew Wade. Wade’s over against New Zealand in Melbourne was considerably less threatening than his spell in Perth, suggesting that he is less effective with the red ball than the pink one. This suggests that going with two spinners is a highly risky move for them.

Another option could be to select Michael Nesser as an all rounder to replace Wade in the side, allowing more cover for the extra spinner, but lengthening the tail considerably. This is unlikely to happen, as Australia have traditionally shied away from picking five bowlers in test sides, and Tim Paine has made it clear that he does not favour changing the formula too much.

New Zealand have taken Todd Astle on a holiday so far, not playing any tests on this tour or in the matches in New Zealand. Will Somerville has been added into the squad, and his familiarity with the conditions and point of difference with his height is likely to make him a tempting option. Somerville is a former Sydney resident, and played for New South Wales for a few years before returning to New Zealand to try to play international cricket. He is close to 2 metres tall, and so created different challenges for batsmen by being able to extract similar bounce to a bowler bowling with loop even while bowling on a flatter trajectory.

Astle provides the advantage of being a competent batsman, so bringing him into the side in place of de Grandhomme is a possibility. That would allow a 3rd genuine seam bowler into the side. Another option is for Astle and Somerville coming in with Tim Southee and Neil Wagner with de Grandhomme acting as the 3rd seamer.

3. Will the pitch actually turn, or is the spinner’s SCG a bit of a myth?

Over the past 10 years, spin bowlers have bowled over 1000 overs at the SCG, but only picked up 82 wickets at an average of over 50 at the SCG. Pace bowlers have taken 192 wickets at and average just under 35 there. Nathan Lyon has averaged 47 at the ground in that time, and collectively the leg spinners used there have averaged roughly 70. The days of Stuart McGill ripping teams apart on the SCG seem to be long gone.

However, when looking at the way that the series has progressed, Australia might consider favouring spin more. New Zealand’s two standing quick bowlers have not been much less effective than their Australian counterparts. Southee and Wagner have taken 26 wickets at less than 23 runs each, while Pat Cummins and Mitchell Starc have taken 19 wickets at just under 18 each. However, Lyon has been much more effective than Santner (10 wickets at 22.7 vs 1 wicket at 250). Giving Lyon slightly more to work with might exaggerate that difference even more.

4. Will New Zealand keep trying to out last Australia with the ball?

New Zealand have had a clear bowling plan in this series. With the new ball: pitch it up, and try to get it to swing occasionally, but mostly bowl a 4th stump line, on a good length. With the older ball, bang it in short of a length. Both tactics have been mostly designed to get the batsmen to play risky shots and get out doing so, rather than trying to actively dismiss the batsmen.

While that tactic has been reasonably successful for Southee and Wagner, it has meant that there has been a lot asked of the other bowlers, and they have not been as successful. Perhaps bowling 1m fuller, and more at the stumps. Cricviz released some interesting data recently that of all batsmen who have faced 500 balls aimed at the stumps since 2006, only Steven Smith averages over 33 against those deliveries, and of players who are still active test batsmen, Virat Kohli has the third best average against balls targeting the stumps of 24.08. That suggests that bowling straighter might be a better tactic. The odd delivery will be hit through the leg side or down the ground, but the approach may well bear more fruit.

The difference in length and line from the Australian bowlers has been clear. They have tended to bowl at the stumps more. Some of that is due to the different styles, but some of it is just that they had different plans, and those plans (especially when a batsman was new to the crease) have been much more effective.

5. Should the match even be going ahead?

Cricket is the job of the players, and of the administrators, but it is still at its heart a game. Is there a point where playing games in the midst of an ongoing natural disaster becomes a little insensitive? Should this match even be going ahead?

The smoke from the New South Wales bushfires has been so thick that the views of mountains in Southern New Zealand (over 2000 km away) has been blocked and some of the New Zealand glaciers have turned brown. In terms of distance, that would be like smoke from a fire in Dubai blocking out the view of the buildings at one end of Marine Drive in Mumbai from the other.

The Remarkables have disappeared. Pics from yesterday and today. 🙁 #AussieSmoke pic.twitter.com/KQjRgwwb5N

— Margie (@Margie186) January 1, 2020

The question has to be asked as to what point is it where player welfare comes to the fore? The atmosphere in Sydney is so polluted from the fires that one lung professor likened breathing it to smoking 40 cigarettes. The PM2.5 reading in some outer suburbs of Sydney was 734. To put that in context the match in Delhi that was called off between India and Sri Lanka had a PM2.5 reading of under 400.

Sport can be important for the morale of people who are experiencing a traumatic event, but there is such a thing as being too soon, and while the bodies of the dead from the fires are still not yet buried it may be too soon to be playing games. Even if the timing is acceptable to the public, is the safety issue to the players too extreme for such triviality.

Statistical preview, World Cup final, New Zealand vs England

Here is a brief statistical preview.

Recent head to head:

In the past 5 years England lead 8-5.
In the past 2 years England lead 6-3.

At Lord's the ball tends to bounce a bit more. As a result it tends to not suit England as much as their other home grounds. It is the only ground that England have a losing record at over the past few years, with 3 wins and 4 losses in their last 6 years.

It is also a ground where scores have been defended quite regularly.

The slope, large straight boundaries and the bounce combine to make a more bowler friendly ground than most in England, but grounds in this world cup have not exactly gone to type.

Adding in times where New Zealand bat first, and where England bowl first, gives the following result:

New Zealand had a clear plan to use the pressure of the situation as a weapon to help them defeat India, and the pressure from playing at home may do the same against England.

The model that I used to build my simulation has England at 69.8%, while New Zealand are at 30.2%. That feels about right too, New Zealand have a realistic chance, but England are certainly favorites.

The bookies have England at 73%, CricViz have England at 68%, New Zealand at 30% and a tie at a fairly high 2%.

The two teams are close enough that nobody can say exactly who will win, but it is a World Cup final - that's exactly how it should be.

World Cup Simulation update - 5 July

Here's the latest update for the world cup simulation. I have New Zealand at 100%, but that's simply due to the probability of Pakistan getting the required run-rate being so low that that possibility never eventuated in the 50000 trials that I used. The probability of Pakistan going through is slightly lower than the probability of someone being shot accidentally by a dog running along a beach while holding a handgun in it's mouth during the next week,

The next graph is the expected points. The simulation has had the correct top 4 from the second match on, however, the expected points and the order of the teams have changed considerably

The top 4 was looking fairly likely from about match number 6 on. There was some excitement from the two upset losses by England, but Pakistan never got beyond 40% on the simulation.

The complete make up of the semi-finalists has not yet been decided, nor has the team in 5th place. Pakistan, Bangladesh and Sri Lanka could all end up 5th. 

Next I looked at the winning probability. This is getting close to the point where it can be calculated analytically without much trouble.

 The next thing to look at is the rankings. A thing to remember here is that it is all relative to Afghanistan, so everybody going up is more an indication that Afghanistan has gone down.

The order that the teams are in here is the same as David Kendix' official rankings order, with one exception - I have India ahead of England, rather than the other way round.

Finally, a little graph to show what Pakistan needs to do to make the semi-finals. They need to keep Bangladesh below the green line.

World Cup simulation update - 1 July

Here's the latest update to the simulation. The first two graphs disagree slightly, and that's because I have two different methods to calculate the expected net run rate. The first one seemed to be slightly more accurate than the second, but there was not a big difference when I tested them. (The margin of victory in cricket matches is actually really difficult to estimate - teams batting second tend to cruise to victory rather than try to win by as big a margin as possible) I decided to use both when doing the calculations. With the first method, New Zealand and India both have a higher than 99.98% probability of going through, while it's 99% for India and 97.7% for New Zealand with the second method. These seem more realistic.

The big thing to notice is the change to England's probability, and how England beating India damaged the chances of both Pakistan and Bangladesh. Pakistan's probability went down by slightly more than Bangladesh's probability because the ranking of India dropped slightly, and Bangladesh need to beat India to get through.

This graph shows expected value - not the most likely value. Those are actually different things. The expected value is the mean of all the expected outcomes. As a result, none of the teams will actually end up with the points that this shows, but they should mostly get close to it.

 It's now looking like there's a roughly 45% chance that net run rate will be a deciding factor in who goes through to the semi-finals.

If Bangladesh beat India (which is admittedly a fairly unlikely outcome), we could then see a situation where Pakistan and Bangladesh are playing for the opportunity to be level on points with New Zealand and India on 11 points. If that is the case, then (in all likelihood) the rained out match between New Zealand and India will have allowed both to progress at the expense of the winner of Pakistan vs Bangladesh.

The most likely semi-finals at this point are Australia vs New Zealand and England vs India, but these are by no means confirmed yet.

In individual matches, England effectively has a higher ranking than that, because teams playing at home get a ranking boost of 0.86 over their opponent. That's why I have England back on top in the next graph:
This one is quite different to what the book-makers have. I have England as favourites, while they have India and Australia both tied for favourite on roughly 30%. They also have Pakistan and Bangladesh at about double the probability that I do.

I used the first net run rate model for the winning probability, but the difference in numbers suggests that the bookies are possibly using a model that is more similar to the second one.

World Cup simulation update - 26 June

Are the wheels falling off?

England have now got a 4 win, 3 loss record, and, with 2 difficult matches coming up, have a genuine chance of not going through to the semi-finals. They are still not relying on other results, but they're getting close to the point where they are.

There's been a significant change, with Australia going up, and England going down. England are now expected to get to 10 points. That might still be enough. But it also might not be.

England's ranking has now dropped well below India's, to the point where the expected probability of England winning against India has dropped by almost 10%. They're still ahead due to home advantage, but the difference is decreasing.

There's about a 15% chance that a tie-breaker (total wins or net run rate) will be required. This may count out Sri Lanka, who have had two rain affected matches, and so will probably be on fewer wins than anyone else with the same number of points.

We see a huge drop in the semi-final probability of England, and a resultant increase in Bangladesh, Pakistan and Sri Lanka. Australia have qualified now, and there are fewer options now for New Zealand to be knocked out also (only 35 out of 50000 trials saw New Zealand miss the semi-finals.)

The decrease in England, and increase in probability of lower ranked teams making the semi-finals has meant that there are a lot more semi-final combinations with more than a 0.5% chance of happening. West Indies vs New Zealand was an epic match in the pool play, and that's now a reasonable possibility for a semi-final. The ICC and Star Sports will be licking their lips at the prospect of the 8th most likely outcome - an India Pakistan semi-final would be absolute ratings gold.

This is the first time that England has dipped below India on the winning probability graph, but it's hard to win the final if you don't get out of the group stage.

World Cup Simulation update 24th June

 Here's the update after the South Africa vs Pakistan match

Firstly, this pushed Pakistan's ranking back above Bangladesh's ranking, although they are both so close that the match between them is now predicted as 50.2% to 49.8%.

 Looking at the expected points, Pakistan have now jumped ahead of Sri Lanka and Bangladesh.

It's looking fairly likely that 5th place will be on 9 or 10 points, while 4th will be on 10, 11 or 12 points.

My simulation only uses net run rate as the tie breaker. Accordingly, there's actually a slightly higher probability of Sri Lanka and Pakistan getting through than this shows, and a slightly lower chance of England and Bangladesh.

It's takes a lot of processor time to improve the simulation, and it's likely to be less than 1% difference, but I might have a go at improving it once we get to the last 5 matches.

England are still the overwhelming favourite to be the 4th team to go through. There were still 41 out of the 50000 trials where New Zealand hadn't made it. So nobody is guaranteed through just yet.

If you have semi-final tickets - this is who you're likely to see.

The probabilities for Bangladesh and Pakistan being so low here are understandable. They both have about a 5% chance of making the semi-final, but, given that they both have about a 1/3 chance of winning each match against the top teams, it gives them a roughly 0.5% chance of winning the tournament from here. However, if Bangladesh, Australia and Pakistan win the next 3 matches, that number will rise.

It's starting to look like England's style that is so effective in series may not be so effective in one off matches. It will be interesting to see if that trend continues.

World Cup Simulation Update, 23 June

Here's the latest outputs from the simulation.

England's loss to Sri Lanka opened the door somewhat, but we can still be fairly confident in who the semi-finalists are.

 England's ranking has gone down, after two losses to fairly ordinary sides.

It's looking like 10 points will be the magic number. Roughly a 10% chance that we'll rely on a tie-breaker.

The average points expected certainly favour England on that count to be in fourth


Accordingly, they have a much higher chance of making it through.

What the likely match ups are. (Teams in alphabetical order, rather than placings)

England are still firm favourites by my model. Home advantage is massive.

World Cup simulation update

The group stage of the World Cup is now roughly half way through, and there are 4 clear favourites to be the semi-finalists.

Afghanistan is the first team to be eliminated (they may have a mathematical possibility, but they don't have a statistical one). At this point, Sri Lanka are not far behind.

The rankings of the teams have remained fairly consistent, suggesting that the extra weighting for world cup matches is about right.

The fact that almost all the teams seem to have gone up is due to them all being relative to Afghanistan. Afghanistan do not seem to be quite as good as they were seeming to be and so they have dropped, but as they are set to 0, it's pushed everyone else up slightly.

The semi-final probability is the most interesting. 

I personally feel that this is underestimating the chances of South Africa, but we will see as the tournament progresses.

The key point on this graph is match 5, where Bangladesh overcame South Africa. If South Africa had won that match, they would be on about 40% and New Zealand and Australia would both be a lot lower.

The simulation also puts out the points for 4th, 5th and the difference between them. This suggests at the moment that there's only a fairly low chance that net run rate will come into play. However, one more rained out match, or a Bangladesh upset of Australia, and this could change dramatically. This makes the expected lines to be 9 points for 5th place, and 11 points for 4th place.

So far of the teams that I've had as favourite to win, 14 out of the 17 have won. Given the probabilities that the models assigned them, that's slightly higher than I would have expected - I would have expected there to have been 4 upsets rather than 3, but it's still telling me that my model is working quite well. That may be due to teams not always playing their best combinations in every match between the world cup, adding extra uncertainty to the results than exist inside a world cup.

It will be interesting to see if it continues to have the same success rate after the cup is finished.

Finally, applying the same system to find the probable winner gets the following results:

England are still favourites, but India are not far behind them.

Preview - World Cup group match 3 - New Zealand vs Sri Lanka

This match is at Sophia Gardens in Cardiff. It's likely to be cool and damp, but with no rain. That's likely to play into New Zealand's hands.

New Zealand are distinct favourites - Bet365 have them at 78%, Google has them at 79%, and my model has New Zealand at 81%. However, none of those are at 100%, and the match isn't played on paper - Sri Lanka are still capable of pulling out a big performance.

Sophia Gardens is an odd shape, similar to Eden Park in Auckland, so it's a shape that New Zealand should be comfortable with. However, New Zealand has a mixed record at the ground - it was host to the match where New Zealand famously lost to Bangladesh in the Champions Trophy. In the one previous match between the two sides there, New Zealand won by 1 wicket, only just managing to win despite bowling Sri Lanka out for 138.

Teams batting first have generally not done well at Sophia Gardens unless they get a very big score. It's likely that both teams will want to chase here.

Again a score of 290 would be below par based on historical data, but ICC events sometimes have the pitches in different conditions to normal matches, so there's a chance that a lower score might still be very competitive.

As with some of the other matches, one of the more interesting things here will be the selections. What combination of players will each team go for?

Whichever way it goes - matches at Cardiff have tended to be interesting, even when the teams have seemed to be mismatched on paper before hand, so this could be the first match that's actually interesting on the field as well as just in the lead up.

A simulation to see who will win the World Cup

One of the main purposes of statistics is to help inform decisions. Cricket statistics are often used when deciding on selection of players, or (more often) arguments about who is the best at a particular aspect. They can help decide which strategies are best, what an equivalent score is in a reduced match (with a particular case of Duckworth Lewis Stern) or which teams should automatically qualify for the World Cup (David Kendix). They are often also used by bookmakers (both the reputable, legal variety and the more dubious underworld version) to set odds about who is going to win.

I decided to attempt to build a model to calculate the probability of each team winning, based on their previous form. This was going to allow me (hopefully) to predict the probabilities of each outcome of the world cup, by using a simulation. It didn’t prove to be as easy as I had hoped.

My first thought was to look at each team’s net run rate in each match, adjust for home advantage, and then average it out. That seemed sensible, and the first attempt at doing that looked like it would be perfect. Most teams (all except Zimbabwe) had roughly symmetrical net run rates, and they fitted a normal curve really well. The only problem was that Afghanistan was miles ahead of everyone else. The fact that they had mostly played lower quality opponents in the past 4 years meant that they had recorded a lot more convincing wins than anyone else.

This was clearly a problem. India and England both had negative net run rates, while Afghanistan, Bangladesh and West Indies were all expected to win most of their matches.

I then tried a different approach, based off David Kendix’s approach of using each result to adjust a ranking. But rather than having a ranking that was based off wins, I based it off net run rate. So if a team had an expected net run rate of 0.5, and another had an expected net run rate of 0.6, the first team would have an expected net run rate of -0.1 for their match. If they did better than that, they went up, and if they did worse than that, they went down.

However, I found that some results ended up having too much bearing. If I made it sensitive to a change in the results, it ended up changing way too much based off one big loss/win. England dropped almost a whole net run per over based on the series in the West Indies. So this was clearly not a good option.

Next, I decided to try using logistic regression, and seeing how that turned out. Logistic regression is a way of determining probabilities of events happening if there are only two outcomes. To do that, I removed every tie or match with no result, and set to work building the models.

My initial results were exciting. By just using the team, opposition and home/away status, I was able to predict the results of the previous three world cups quite accurately using the data from the preceding 4 years. (I could not go back further than that, as they included teams making their ODI debut, and there was accordingly no data to use to build the model.

The results were really pleasing. I graphed them here, grouped to the nearest 0.2 (ie the point at 0.6 represents all matches that the model gave between 0.5 and 0.7 as the chance for a team to win), compared to the actual result for that match. It seems that they slightly overstate the chance of an upset (possibly due to upsets being more common outside world cups, where players tend to be rested against smaller nations), but overall they were fairly reliable, and (most importantly) the team that the model predicted would win, generally won.

I could then use this to give a ranking of each team that directly related to their likelihood of winning against each other. The model gave everything in relation to Afghanistan, with the being 0, and any number higher than 0 being how much more likely a team was to win against the same opponent as Afghanistan. (Afghanistan was the reference simply because they were first in the alphabet).

This turns out to be fairly close to the ICC rankings. So that was encouraging.

I tried adding a number of things to the model (ground types, continents, interactions, weighting the more recent matches more highly) but the added complexity did not result in better predictions when I tested them, so I stuck to a fairly simple model, only really controlling for home advantage.

Next I applied the probabilities to every match and found the probabilities of each team making the semi-finals.

The next step was to then extend the simulation past the group stage, and find the winner.

After running through the simulation a few more times, I came out with this:

A couple of points to remember here: every simulation is an estimate. The model is almost certainly going to estimate the probabilities incorrectly, but it will get them close, and they will be close enough to give a good estimate of the actual final probabilities. It is also likely to overstate Bangladesh’s ability due to their incredible home record; overstate Pakistan’s ability as a lot of neutral matches for them they have had a degree of home advantage in UAE; and understate West Indies, due to them having not played their best players in a lot of matches in the past 4 years. But these are not likely to make a massive difference to the semi-finalist predictions.

Given this, I’d suggest that if you are wanting to bet on the winner of the world cup, these are the odds that I would consider fair for each team:

I will try to update these probabilities periodically throughout the world cup, and report on their accuracy.

Historical statistical preview of the Second Test, Pakistan vs NZ

I've decided to put together a short summary of some of the historical trends at Dubai, before this match.

First, the probability of different results based on first innings scores. This suggests that a score of 300 is roughly the point where a team is more likely to win than lose, while the 50% winning score is roughly 370.

Plunket Shield update - Round 2

At the end of round 2, I thought it would be good to do an update on the progress of the tournament, and look at some trends that have emerged.

One thing that I thought I would focus on is how the runs have been scored, rather than just how many.

I looked at each innings and looked at the total runs from boundaries, and the total other runs (I called them run runs, but they include no balls and wides, as they were too hard to separate).

I plotted them on a graph, to see if there were any interesting patterns emerge.

There were a couple of things that I noticed. Auckland, Otago, Wellington and Canterbury have all had similar rates across the different innings that they've batted, while Northern Districts and Central Districts have had more variety in how they've accumulated their runs.

The triangles seemed to be higher up the chart on average, with all of them being above the median boundary rate, so I thought that I'd see if there was a correlation between the rates and the total competition points gathered in a match.

There is a reasonably strong relationship between the boundary rate and the points earned in a match, however, there's almost no relationship at all between the speed of accumulation of non-boundary runs and the points earned.

There is a theory that regularly rotating the strike makes it easier to survive a match, as it doesn't allow the bowlers to settle. I certainly know that I hated batsmen hitting singles off my bowling, and I remember Dale Steyn saying in a press conference something to the effect of "I don't mind dropped catches that much. Dropped catches happen. But I get really upset when a fielder lets a batsman get off strike when I had him under pressure."

Of the 7 innings played by a losing side, 5 of them had a run-runs rate below the median. That made me wonder if there was a pattern there. I looked at the final innings by teams that batted out a draw or lost, and looked to see if there was a difference in the rates for the teams that lost vs the teams that drew.

This graph isn't particularly meaningful at the moment, with only 5 innings to look at, but I intend on building this up as the season goes on.

Looking at it as individual points makes it more clear:

I've circled the point at the bottom, because that was an innings where Canterbury lost their last wicket with only 6 balls remaining, and so it was very close to being a saved match. Interestingly the teams that have scored a lot of boundaries have lost, but it is a very small sample to be drawing too many conclusions from.

The final table, with other information, looks like this:

This made me wonder which correlation was stronger, scoring rate with total points, or the traditional value of Net Average Runs Per Wicket (batting average minus bowling average).

The Net Average Runs Per Wicket seems to be a better predictor of success, but there is a clear relationship with the scoring rate also.

I'll be interested to see how these develop as the season progresses, but for now we seem to have a separation between the sides, with Auckland, Canterbury and Otago all needing to find another gear for the next round.