# Are wickets more likely on hat-trick balls?

A student of mine has emailed me asking if I know anything about whether

in cricket a wicket is more or less likely on a hat-trick ball then on any

other ball. (Note for Eric and others similarly challenged in the finer nuances

of cricket, a hat-trick occurs when a bowler takes a wicket with each of three

consecutive deliveries. Note for followers of other sports: this is the original use of the term “hat-trick” in sport.) The student and

his flatmate have surmised that taking a wicket on a hat-trick ball is more

likely than on any other randomly chosen ball. I don’t know what the data say

on this, but I think the students are almost certainly right, mostly for

statistical reasons. It is fun to think about how to formalise the hypothesis,

and then how to test the effect of different forces. Maybe it could be a future

Honours project to take this theory to the data.

in cricket a wicket is more or less likely on a hat-trick ball then on any

other ball. (Note for Eric and others similarly challenged in the finer nuances

of cricket, a hat-trick occurs when a bowler takes a wicket with each of three

consecutive deliveries. Note for followers of other sports: this is the original use of the term “hat-trick” in sport.) The student and

his flatmate have surmised that taking a wicket on a hat-trick ball is more

likely than on any other randomly chosen ball. I don’t know what the data say

on this, but I think the students are almost certainly right, mostly for

statistical reasons. It is fun to think about how to formalise the hypothesis,

and then how to test the effect of different forces. Maybe it could be a future

Honours project to take this theory to the data.

Take a set of games in a particular format (say test cricket), and find

the total number of deliveries and the fraction of those that resulted in the

bowler being credited with a wicket. Then find the total number of deliveries

in all those matches where, if the bowler had taken a wicket he would have achieved

a hat-trick, and find the fraction of those deliveries where a wicket was in

fact taken. Our guess is that this latter fraction will be higher than the general

fraction of deliveries with wickets, and that that difference would be statistically

significant. I am fairly confident about this purely because of sample

selection:

the total number of deliveries and the fraction of those that resulted in the

bowler being credited with a wicket. Then find the total number of deliveries

in all those matches where, if the bowler had taken a wicket he would have achieved

a hat-trick, and find the fraction of those deliveries where a wicket was in

fact taken. Our guess is that this latter fraction will be higher than the general

fraction of deliveries with wickets, and that that difference would be statistically

significant. I am fairly confident about this purely because of sample

selection:

- Pitches vary considerably across matches; if a bowler has already taken

two wickets in two balls, it is likely that the pitch for that game (and that

point in the game) is an easier one for taking wickets than the average. - Bowlers (and their supporting fielders) vary in ability; if a bowler has

already taken two wickets in two balls it is likely that he is a better bowler (with

better supporting fielders) than the average. - Batsmen vary in ability and batter ability is both correlated within the

batting order and correlated within teams; if a bowler has taken two wickets in

two balls it is likely that the batting team has below average quality batsmen

and that it is one of the weaker batsmen in the team who is facing the

hat-trick ball. - Statistically (I can confirm this from test-cricket data), batsmen are

more at risk at being dismissed early in their innings than later on; there is

a high likelihood that the batsmen facing the hat-trick ball is facing his

first ball of the innings.

So let’s control for these sample selection issues and consider instead

a conditional probability question:

situation leads players to change their behaviour in some way. On the bowling

side, the captain might set more aggressive wicket-taking fields on a hat-trick

ball, but the bowler might try too hard and lose his rhythm. Similarly, the

batsmen might be more conscious about not giving his wicket away, but at the same

time the pressure of the situation might lead to his having leaden feet.

a conditional probability question:

*Given*

the ability of the bowler and fielders, the batsman, how early it is in the

batsman’s innings and the state of the pitch, does being on a hat-trick change the

probability of a wicket?The question here becomes whether the unusualthe ability of the bowler and fielders, the batsman, how early it is in the

batsman’s innings and the state of the pitch, does being on a hat-trick change the

probability of a wicket?

situation leads players to change their behaviour in some way. On the bowling

side, the captain might set more aggressive wicket-taking fields on a hat-trick

ball, but the bowler might try too hard and lose his rhythm. Similarly, the

batsmen might be more conscious about not giving his wicket away, but at the same

time the pressure of the situation might lead to his having leaden feet.

I would expect that the psychological effect would be greater on a

batsman new to the crease than a bowler who has had a chance to find his

rhythm. And in test cricket, I think that batsmen are always concentrating only

on wicket preservation on the first ball they face. So If I had to guess, I

would say that in test cricket the net result would still be that wickets are

more likely on balls where the bowler is on a hat-trick, but the effect would

be very small (and probably not discernible with statistical significance in

the data). In limited overs cricket, I would expect the effect to be much smaller

or even zero.

batsman new to the crease than a bowler who has had a chance to find his

rhythm. And in test cricket, I think that batsmen are always concentrating only

on wicket preservation on the first ball they face. So If I had to guess, I

would say that in test cricket the net result would still be that wickets are

more likely on balls where the bowler is on a hat-trick, but the effect would

be very small (and probably not discernible with statistical significance in

the data). In limited overs cricket, I would expect the effect to be much smaller

or even zero.

Now, if only I had ball-by-ball data for the entire history of test

cricket!

cricket!

## 2 Responses to “Are wickets more likely on hat-trick balls?”

What about regression to the mean?

Regression to the mean would explain why the probability of a bowler taking a wicket on the hat-trick ball would be less than the fraction of balls in the match so far on which he has taken a wicket. RTM could potentially be a factor when constructing the conditional probability of a wicket on the next ball, to find a null, deviations from which would be the measure of psychological effects of being on a hat-trick. That is, failure to take into account RTM could lead us to falsely infer that the psycholgoical effects are such to make a wicket less likely than normal. That is, RTM won’t be a reason to think the conditional probability question should go one way or the other, but it might be a reason for us to falsely infer that it does.