# **PROBABILITY SPACES IN TEST CRICKET**

The first Test at Abu Dhabi is moving on quickly. You have just checked WinViz and it has Pakistan at 42%, England at 35%, and the draw at 23%. You then look at PredictViz and it is showing an England win by a handful of runs. Surely something has gone wrong? One tool is saying Pakistan are going to win, the other that England are.

There has not been a technical meltdown; the CricViz tools are working correctly. Remember, we are not predicting a Pakistan victory. In fact, we think Pakistan are more likely to not win (58%), than win (42%). Therefore, it should not be a huge surprise that the average result is not a Pakistani win.

But wouldn’t you expect the average result to come from the most likely outcome?

Well, sometimes it does, sometimes it doesn’t. There are a couple of concepts to understand before looking at the details of why this happens.

First of all, percentiles: If you are in the 40th percentile for height, you would expect 40 out of 100 people to be your height or shorter. If you were in the 95th percentile for height then you would expect only five in every 100 people to be taller than you.

Similarly, if we look at the possible range of scores for a Test team we can use percentiles to measure how likely a team is to make a certain score. For example, let’s look at an average team batting second in a Test match. A score of 420 is in their 70th percentile. 30% of the time they will score more than 420.

Secondly, let’s take a look at probability spaces. Imagine we each roll a six-sided die and add the two numbers together. There are 36, equally likely outcomes, which we can illustrate in a diagram:

As you can see, the total eight occurs five times, so we can say that the probability of getting a total of eight is five in every 36, or 5/36.

We are now going to play a game. The rules are as follows:

1.) If your score is greater than or equal to mine, then you win

2.) If my score is greater than yours, I win.

3.) However, if we both throw a 4 or higher, then it is a draw.

How likely is each of the three results? We can look at the probability space to tell us:

So, in 15 of the 36 possibilities you win, so you will win 5/12 of the time. I will win 12 out of 36 or a third of the games, and a quarter will be draws.

We can take the same approach with Test matches, by creating a probability space based on how well each team scores during the match. Let’s take a look at the probability space of a Test match between two well-matched teams with no weather interruptions.

Here the batting performances of each team form our axes. So the left-hand side are low Team A scores, the right-hand side are high scores. The top half of the chart shows good Team B batting performances, the bottom half are poor performances. As you can see, when both sides score highly (top, right-hand corner) we get a draw. When one side or the other scores poorly, they tend to lose.

#### Probability space for two well-matched teams:

So, if Team A’s scores are in the 70^{th} percentile then we are looking at the column above the number 70. And we can see by looking up this column that they won’t lose if they bat this well. The result now depends on Team B’s batting. If they perform better than their 40^{th} percentile then they will save the match, otherwise Team A will win.

One thing you will notice is that the chart is not exactly symmetrical. The team batting first has a slight advantage in terms of scoring; the pitch tends to be more batsman friendly in innings 1 and 3 than in 2 and 4. So in low-scoring matches, where there will be a result, Team A has a slight edge. If you look at the 100 squares (from 5^{th} to 50^{th} percentile for each team) in the bottom left-hand section of the chart, you will see that 53 of them are blue Team A wins, and 47 are red Team B wins. The blue squares include the 50^{th} percentile match, our median match, which is what PredictViz shows. If both teams produce their average performance, then the result will be a Team A win by a very small margin. This is true even though the balance of power lies with Team B who will win 7% more matches than Team A.

So, if Team A scores more runs, why does Team B win more matches? Well, the key is what happens in relatively high-scoring games. It is far easier for Team B to force a result in matches where they bat well. Look at the top, right-hand corner of the chart where both sides have batted better than average. You can see that the red squares extend into this space where as the blue squares don’t. There are no wins to the team batting first in matches where both sides score relatively highly.

We can see what this means in reality by thinking about the effect of a first innings lead. If Team A gets a first innings lead, then to win the match they will generally bat on until they can make the game safe, declare and then bowl out the opposition, who know just what they have to do to save the game. On the other hand, if Team B get a first innings lead, they just have to bowl Team A out and chase down the resulting total. They are able to use the time left in the match far more efficiently to force a result.

PredictViz shows what will happen if both teams perform exactly as expected, how their innings will fall across the days of the match and what the average shape of the match from here will look like. It is obviously showing just one of the infinite ways that the match can evolve from here. How many of those fall to each team is represented by WinViz.

**PROBABILITY SPACES IN TEST CRICKET**

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