# Are wickets more likely on hat-trick balls?

By Seamus Hogan 24/10/2013

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.

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:
• 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: 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 unusual
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.

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

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

• JLM says:

What about regression to the mean?

• Seamus Hogan says:

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.