Chief Scientist of Australia » mathematics http://www.chiefscientist.gov.au Mon, 30 Jan 2012 22:42:35 +0000 http://wordpress.org/?v=2.8.4 en hourly 1 Maths in real life: Horses for (maths) courses http://www.chiefscientist.gov.au/2011/05/maths-in-real-life-horses-for-maths-courses/ http://www.chiefscientist.gov.au/2011/05/maths-in-real-life-horses-for-maths-courses/#comments Fri, 20 May 2011 06:51:56 +0000 ACooper http://www.chiefscientist.gov.au/?p=3422 post thumbnail

Naomi White is a student at Box Hill TAFE in NSW, learning about Equine Studies. Here, she outlines why maths is so important to caring for horses.

“There is maths in feeding a horse. Feed requirements and medication doses are based on the horse’s weight.

“Estimating the weight of a horse is a maths equation – it is approximately equal to the girth measurement squared, multiplied by the length of the horse from the chest to the tailbone, divided by 12,000.

“The basic feeding requirement is that you have to give the horse sufficient energy to maintain itself in the paddock.  We know that a horse can only consume 2.5% of its body weight in food every day.  Then we use ratios to make sure the mix of nutrients provides the horse with its daily requirements.”

Likewise, Manager of the Glenbrae Equestian Centre, Brian Hodgson emphasises the importance of mathematical knowledge in his job.

“I’m the owner manager of Glenbrae Equestrian Centre, where we agist horses, run vocational training programs, conduct riding lessons and run competitions.  There is a great deal of mathematics involved in the things that I do on a day-to-day basis.

“The arenas for dressage competition have to be a rectangle 20 metres by 60 metres. We use Pythagoras’ Theorem to make sure we have the lengths and angles correct.  The horse’s stride length determines the distance between jumps for competition. We use spreadsheets to allocate competitors their position in the competitions and use statistical analysis for scoring.  I’ve found mathematics to be invaluable throughout my life.

“I had no idea when I was going through school how useful maths would be. It has opened up doors for me.  The further you pursue mathematics the more opportunities that are going to be open up to you in the equine industry.”

Profiles of Naomi and Brian, including videos and posters are available on the Maths Careers page, courtesy of the Australian Mathematical Sciences Institute, ICE-EM Mathematics and the Australian Government’s Department of Education, Employment and Workplace Relations.

]]> http://www.chiefscientist.gov.au/2011/05/maths-in-real-life-horses-for-maths-courses/feed/ 0 The birthday problem http://www.chiefscientist.gov.au/2011/05/the-birthday-problem/ http://www.chiefscientist.gov.au/2011/05/the-birthday-problem/#comments Thu, 19 May 2011 06:33:51 +0000 ACooper http://www.chiefscientist.gov.au/?p=3410 post thumbnail

Say you work in an office with around 20 employees. If two of them shared the same birthday, would you remark on it as an interesting coincidence? A rarity?

Many people might, because our instinct tells us that it’s quite unlikely that two people in an office of around 20 people were born on the same day of the year. After all, there are only 20-odd people and there are 365 possible birthdays (for simplicity’s sake, let’s ignore the possibility that anyone was born on February 29 in a leap year).

But, are our instincts right? Mathematics allows us to work out just how likely, or probable, such a coincidence is.

The answer might surprise you. Take an office with 23 employees: with 23 randomly-selected birthdays, the probability that two or more will be the same is just over 50 per cent—a better than even chance! And with those odds, it’s hardly remarkable when such a coincidence occurs.

Read on if you’d like to get into the nitty-gritty of the probability calculation.

In the mean time here are some other results to variations on the ‘birthday problem’: in a group of 40 randomly selected people, the chance that two or more would share the same birthday goes up to around 90 per cent; in a group of 14 people, the chance that two or more would have birthdays that are either the same or only one day apart is around 50 per cent; in a group of 88 people, there’s an even chance that three or more people would share the same birthday.

So, how do we work this out? By working out the probability that there are no shared birthdays in the group, we can then work out the probability that there are two or more people sharing a birthday.

This is because these are the only two possible outcomes—either (a) no one in the group shares a birthday with another in the group, or (b) at least two people share the same birthday. Because these are the only two possibilities, their respective chances must add up to 100 per cent, and by working out one chance, we can work out the other.

Let’s put our group of 23 people in a line and work our way through from end to end.

The first person will have one of 365 possible birthdays (as described below*). Now we work out the probability that the second person in line does not share the same birthday as the first, which is 364/365. Why this number? Because, out of the 365 possible birthdays for person number two, 364 of them will not match the first person’s birthday, thus giving a probability of 364/365.

Next we take the third person in line and work out the probability that he or she does not share a birthday with either of the first two: this chance is 363/365, following the reasoning of there being 363 out of 365 possible birthdays that do not match the two birthdays of persons one and two.

Applying the same logic to the fourth person, we get a probability of 362/365 that their birthday will not be the same as the three people before them in the line. We then continue assigning probabilities down the line until we get to the 23rd person, who will have a 343/365 chance of not sharing a birthday with the 22 other people in the group.

We now use the calculated chances that each successive person in line will not share a birthday with those preceding to work out the probability that the entire group has no shared birthdays.

In words, this is the chance that i) the second person does not share a birthday with first (chance = 364/365), AND ii) the third person does not share a birthday with the first or second (363/365), AND iii) the fourth…(and so on)…AND, finally, xxii) the 23rd does not share a birthday with any of the previous 22 (343/365).

The probability that ALL of these things will occur is obtained by multiplying together the chances of each individual event occurring:

Probability that there are no shared birthdays in a group of 23

= 364/365 x 363/365 x 362/365 x 361/365 x ……. x 344/365 x 343/365 (you can work this out on a calculator)

= 0.4927… = 49.3% (rounded)

Therefore, we know the probability that at least two people share the same birthday is:

100% – 49.3% = 50.7% —better than even!

 

*Assumptions behind the calculation: we assume birthdays are spread evenly throughout the year—that is, each day of the year is equally likely as a birthday. In real life this may not be true, as the birth rate can change month to month, season to season, leading to an uneven distribution of birthdays. Some births are more likely to occur on weekdays than weekends, such as those by caesarean section, and this may affect the distribution of birthdays for certain populations (e.g. a class of children all born in the same year).

The second assumption is that we ignore the possibility of people being born on February 29 and examine the case where a population has their birthdays evenly distributed across the 365 days of a non-leap year. Again, this is not true in reality, but considering that February 29 is just one possible birthday out of every four years—so 1 possible birthday out of 1461 days—the simplified case of 365 possible birthdays is accurate enough to give us a general idea about the probabilities involved in the birthday problem.


Further reading:

Matthews, R. and Stones, F. (1998). Coincidences: the truth is out there. Teaching Statistics, 20: 17–19. Available at http://ts.rsscse.org.uk/gtb/matthews.pdf

Crilly, T. (2007). 50 mathematical ideas you really needs to know. Quercus Publishing Plc, London

Written by Sarah White, Science Research Officer in the Office of the Chief Scientist.

]]> http://www.chiefscientist.gov.au/2011/05/the-birthday-problem/feed/ 0 Do the sums: Be a champion http://www.chiefscientist.gov.au/2010/04/do-the-sums-be-a-champion/ http://www.chiefscientist.gov.au/2010/04/do-the-sums-be-a-champion/#comments Wed, 28 Apr 2010 00:09:48 +0000 ACooper http://www.chiefscientist.gov.au/?p=1736 post thumbnail

At the annual  APSM Olympiads, students use maths to battle it out for the chance to help their school take the title of ‘Team of the Year,’ as well as working to gain a perfect individual score for themselves.

Students from any Australian or New Zealand school can compete in one of two divisions – Division J for children under 12 years of age, and Division S for children under 14 years of age. Last year, more than 32 000 students competed in the challenge and organisers are expecting even more for the 2010 event,  with registration due to close on April 30.

Schools can enter teams of up to 30 students and both individual and team prizes are up for grabs.

Former Chief Scientist for Australia, Professor Penny D Sackett, said events such as the Olympiads encourage children to engage with maths in a positive way outside the classroom.

“Maths can be fun, exciting and interesting, and the Olympiad is just one example of all three.

“Australia is facing a skills shortage in many areas that are dependent on knowledge of mathematics – everything from engineering to economics – so it’s important that we encourage students to improve their maths skills,” Professor Sackett said.

Enrolments in secondary school and university maths programs are slipping every year, with the starkest drop recorded between 2005 and 2007 when the number of year 12 students enrolled in advanced mathematics courses declined by 20 per cent.

A recent paper released by the Group of Eight universities which discussed the state of mathematics in Australia, indicated that per capita, Australia trains two and one-half times fewer mathematics graduates than is the standard in developed (OECD) countries.

One of the recommendations of the report is a push to make maths fun and exciting for students.

As well as activities such as the Olympiads, students can become involved in numerous other initiatives and competitions, including science poetry competitions, wildlife photography contests, drawing and video contests and even robot creations.

 To find out more about these competitions, please visit the websites listed below:


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