Category Archives: Mathematics

Science improves the penalty shootout

Mathematics, Psychology

Did you know that sporting clubs are using scientific methods to increase their success rates? After all, success in professional sport is big business ($ millions).

This study analysed video replays of 361 kicks in 37 penalty shootouts at Soccer World Cups to identify predictable behaviours of kickers and goalkeepers.

For each kick, there was an equal chance of the kicker kicking left or right, the goal keeper diving left or right, and the keeper diving in the correct direction (i.e. same as chance).

However, if kickers repeatedly kicked in the same direction, the keeper was more likely to dive in the opposite direction.

This is known as ‘gambler’s fallacy’; the incorrect belief that a run of kicks in the same direction increases the likelihood of the next kick going in the opposite direction.

Rather, it is always a 50:50 chance, independent of previous results.

Kickers should act as a team to exploit this belief and increase their scoring success in penalty shootouts.

 

Do you want more information?

Background

Scientific methods and data analysis can be applied to multiple aspects of life to improve outcomes.

Improving success of professional sporting teams is big business.

For example, a missed penalty by Louis Saha in Manchester United’s loss to Celtic in the 2006 FIFA Champions League is estimated to have cost the club 15 million pounds in lost revenue.

Therefore, sports are using scientific methods to maximise success.

Materials and Methods

This study analysed video replays of 361 kicks in 37 penalty shootouts at World Cups between 1976 and 2012 (36 year period). They recorded the direction of the kick and direction the goal keeper dived.

It was assumed the keeper does not have enough time to observe and react to the kick, but instead anticipates (guesses) which way to dive before the ball is kicked.

Results

For each kick, there was an equal chance of the kicker kicking left or right, the goal keeper diving left or right, and the keeper diving in the correct direction (i.e. same as chance).

However, if kickers repeatedly kicked in the same direction, the keeper was more likely to dive in the opposite direction.

This is known as ‘gambler’s fallacy’; the incorrect belief that in a random binary event, a run of the same result will increase the likelihood of the opposite result (e.g. 3 coin tosses of heads in a row increases the likelihood of the next being a tail).

Instead, each event has an equal chance (50:50), independent of the previous result.

Kickers on the other hand do not display ‘gambler’s fallacy’.

Discussion

Goal keepers are more predictable than the kickers (gambler’s fallacy).

The kickers currently behave as individuals, but if they took a team approach to exploit the keepers belief in ‘gambler’s fallacy’, this could increase scoring goals in penalty shootouts.

It is already tough being a goal keeper. There are multiple kickers in a penalty shootout but only one keeper and the kickers usually score. This knowledge would just make it worse for them.

Article

Asymmetric predictability and cognitive competition in football penalty shootouts

Misirlisoy and Haggard, 2014 Current Biology 24:1918-22

Keywords

Mathematics, statistics, data, analysis, sport, football, soccer, goal, keeper, kick, kicker, video, sport, chance, gambler

Subject

Science, Mathematics, Statistics, ST2-15I, SC4-15LW, ACSHE119, ACSHE134, SC5-15LW, ACSSU184

What makes bird flocks so flocking big?

Biology, Mathematics

Do you know why flocks of birds are big, but not too big?

This study used mathematical modelling to investigate flock size.

Birds form large flocks to protect themselves from predators, but they limit the size of the flock to a maximum density at which individual birds can still see out of the flock.

This means there are more pairs of eyes looking for predators and they could all respond faster to a threat (at the speed of light).

Therefore, there is a balance between flocks being big enough to provide protection and being small enough to still see out of it.

 

Do you want more information?

Background

Small animals often organise themselves into much larger groups, including birds, fish, insects and even mammal herds.

They do this to protect themselves from predators. It provides:

1) Collective awareness: many eyes are better at detecting predators.

2) Confuses predators: many animals packed together makes it difficult

for predators to distinguish individual targets.

3) Presence of many animals reduces the chances of being eaten.

So why aren’t the flocks even bigger? What limits their size?

Materials and Methods

This study performed mathematical modelling to predict the behaviour of individual birds and the collective flock.

They started with the question “What does a bird actually see when it is part of a large flock?”

They measured the opacity of flocks (i.e. light passing through it).

Results

Birds respond to their nearest neighbours. That is, if the nearby birds turn left, then so will that bird.

However, if threatened by a predator, this wave of reaction would be too slow.

Instead, this study shows the size of the flock is limited by the ability of individual birds to see out of it.

This means there are more pairs of eyes looking for predators and they could all respond faster to a threat (at the speed of light).

Discussion

Birds form large flocks to protect themselves from predators, but they limit the size of the flock to a maximum density at which individual birds can still see out of the flock.

Therefore, there is a balance between being big enough to provide protection and being small enough to still see out of it.

Article

Role of projection in the control of bird flocks

Pearce et al., 2014 Proc. Natl. Acad. Sci. USA 111:10422-6

Keywords

Bird, flock, swarm, predator, prey, protection, ecosystem, ecology, animal, hunting, food chain, food web, adaptation, fly, flight, vision, sight, mathematics, modelling

Subject

Science, Biology, Zoology, ST2-11LW, ACSSU072, ST3-11LW, ACSSU094, SC4-15LW, ACSSU112, SC5-14LW, ACSSU175