## Implementation of Umpire's Call for LBW Further Makes Cricket More of a Batsman's Game

*To access the full model, my code and more detailed discussion click on "Download as PDF" above. If you have MATLAB and would like to run further tests using my model, click on "Download Source".*

This is a short statistical study of the inherent bias in the implementation of Umpire's call in DRS for LBW. I compute the probability distribution function(PDF) of the impact of a ball predicted by hawk eye ball tracking on the stumps and infer from it the probability of the ball hitting or missing the stumps in each of the scenarios: hitting, umpire's call and missing. My findings indicate that Umpire's call in the case of LBW review is not actually being used in a 50-50 type situation(as the public and many commentators generally believe) and that the current implementation of Umpire's call is excessively biased towards the batsmen. A potential fix discussed in section 5 is to change the null hypothesis from a batsman being "not out" to the Umpire's call.

Figure: The off side and leg side yellow lines represent the farthest edges of the stumps. The middle yellow line goes through the middle of middle stump. The inner circle represents the size of the ball and the area within which 99.97% of all possible ball centers would be found for a particular scenario. The colors inside the inner circle represent the probability of the ball center being there. The outer circle shows the area of influence which would be covered if you place balls at all possible ball center locations in the inner circle.

## The Bias in Umpire's Call for LBW

This excercise busts one of the most common myths about Umpire's call. Umpire's Call is not being used for 50-50 calls. It is actually being used when the DRS system is over 50% certain that the ball is hitting the stumps. Whereas, even if the DRS system is 51% certain that the ball is missing the stumps, the Umpire's call of out will be overturned. This is completely biased towards batsmen. In the current system, there will be an overwhelming number of cases where Umpire's call is relied upon even when DRS is more than 90% certain that the ball is hitting the stumps (even with a large estimate of uncertainty used = half of ball width). If we want to keep this kind of an implementation, then the DRS system should be more than 90% certain that the ball is going to miss the stumps in order to overturn the Umpire's decision. As it stands, this is currently not the case. Even a 51% possibility of the ball missing the stumps can overturn the Umpire's call if it is out. Therein lies the bias against the bowling team.

Umpires Call Probability (hitting) Result

Out 50-100% Out

Out 0 - 50% Not Out

Not out 50-100% Not Out

Not out 0-50% Not Out

Out 50-100% Out

Out 0 - 50% Not Out

Not out 50-100% Not Out

Not out 0-50% Not Out

## Potential Fix?

In the current implementation of DRS for LBW, the null hypothesis is that the batsman is not out. Therefore, even if DRS says there is over a 90% chance of the ball hitting the stumps (conservative estimate, the chance may be much higher if you believe the accuracy quoted by the ball tracking software) but the Umpire says not out, it stays not out. Whereas a 51% chance of the ball missing the stumps (when the predicted ball path is just grazing the stumps) is enough to overturn a decision of "out".

The null hypothesis could be changed to the Umpire's call. This would truly mean that the Umpire's call will come into play in situations close to 50-50. Moreover, the criteria need to be the same whether the ball is hitting or missing the stumps in order to void the null hypothesis and overturn the Umpire's call. This is not something new, the Umpire's call is already the null hypothesis when a catch(grassed or not) is reviewed and a soft signal is given. The same could be done for LBW instead of biasing it away from the bowling team.

The exact number or thresholds can be worked out by the ICC officials and people who actually are making money from this technology (again, I am using my personal time and resources to do this), but a potential solution will look something like this:

The null hypothesis could be changed to the Umpire's call. This would truly mean that the Umpire's call will come into play in situations close to 50-50. Moreover, the criteria need to be the same whether the ball is hitting or missing the stumps in order to void the null hypothesis and overturn the Umpire's call. This is not something new, the Umpire's call is already the null hypothesis when a catch(grassed or not) is reviewed and a soft signal is given. The same could be done for LBW instead of biasing it away from the bowling team.

The exact number or thresholds can be worked out by the ICC officials and people who actually are making money from this technology (again, I am using my personal time and resources to do this), but a potential solution will look something like this: