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The Gambler’s Fallacy

The above video is an excellent introduction to the gambler’s fallacy.  This is the misconception that prior outcomes will have an effect on subsequent independent events.  The classic example for this is the gambler who watches a run of 9 blacks on a roulette wheel with only red and black, and rushes to place all his money on red.  He is sure that red must come up – after all the probability of a run of 10 blacks in a row is 1/1024.  However, because the prior outcomes have no influence on the next spin actually the probability remains at 1/2.

Maths is integral to all forms of gambling – the bookmakers and casino owners work out the Expected Value (EV) for every bet that a gambler makes.  In a purely fair game where both outcome was equally likely (like tossing a coin) the EV would be 0.  If you were betting on the toss of a coin, the over the long run you would expect to win nothing and lose nothing.  On a game like roulette with 18 red, 18 black and 2 green, we can work out the EV as follows:

\$1 x 18/38 represents our expected winnings
-\$1 x 20/38 represents our expected losses.

Therefore the strategy of always betting \$1 on red has an EV of -2/38.  This means that on average we would expect to lose about 5% of our money every stake.

Expected value can be used by gamblers to work out which games are most balanced in their favour – and in games of skill like poker, top players will have positive EV from every hand.  Blackjack players can achieve positive EV by counting cards (not allowed in casinos) – and so casino bosses will actually monitor the long term fortunes of players to see who may be using this technique.

Understanding expected value also helps maximise winnings.  Say 2 people both enter the lottery – one chooses 1,2,3,4,5,6 and the other a randomly chosen combination.  Both tickets have exactly the same probability of winning (about 1 in 14 million in the UK) – but both have very different EV.  The randomly chosen combination will likely be the only such combination chosen – whereas a staggering 10,000 people choose 1,2,3,4,5,6 each week.  So whilst both tickets are equally likely to win, the random combination still has an EV 10,000 times higher than the consecutive numbers.

Incidentally it’s worth watching Derren Brown (above). Filmed under controlled conditions with no camera trickery he is still able to toss a coin 10 times and get heads each time.  The question is, how is this possible?  The answer – that this short clip was taken from 9 hours of solid filming is quite illuminating about our susceptibility to be manipulated with probability and statistics.  This  particular technique is called data mining (where multiple trials are conducted and then only a small portion of those trials are honed in on to show patterns) and is an easy statistical manipulation of scientific and medical investigations.

If you liked this post you might also like:

Does it Pay to be Nice? Game Theory and Evolution. How understanding mathematics helps us understand human behaviour

Premier League Finances – Debt and Wages. An investigation into the finances of Premier League clubs.

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

Western music has its roots in the harmonics discovered by Pythagoras – himself a keen musician – over 2000 years ago.  Pythagoras noticed that certain string ratios would produce sounds that were in harmony with each other.   The simplest example is illustrated above with an electric guitar.  When a string is played, and then that same string pressed half-way along its length (in the guitar’s case the 12th fret), then we get the same note – this is a whole octave.

If you were to then half the distance again you would get another octave (which explains why guitar frets get smaller and smaller near the base of the instrument – the frets mark ratios relative to the whole string).

The ratio 1: 1/2 shows the ratio of an octave.  A full length string: half length string.  We can multiply both sides by 2 to remove the fraction to get, 2: 1.  This is the octave ratio.

All the other harmonies that are the basis of Western music can also be understood through similar ratios.  The chord sequence E, A, B – which is the standard progression for blues and modern music comprises of the base note (in this case E), along with the perfect fourth (A) and the major fifth (B) of the base note.

Looking at the guitar fret we can see that the perfect fourth (A), which occurs on the fifth fret, has the ratio 1: 3/4.  That is 1 whole string: 3/4 of the whole string.  We can simplify this to get 4:3.

The major fifth (B) which occurs on the seventh fret has the ratio 1: 2/3 which simplifies to 3:2.

The other most likely note used in the key of E would then also be either G (the minor third) which has a ratio of 6:5, or G sharp (the major third) which has a ratio of 5:4).

It’s interesting that we find these particular whole number ratios pleasing to listen to – indeed these are the notes that often sound “right” when playing through songs.  It’s also helpful to look at the circle of fifths – which shows all notes which are in the ratio 3:2 with each other.  Moving around the circle again produces music which sounds nice.  For an example of this, starting at C, the progression C,G,D,A,E is the one used by Jimmy Hendrix in the classic song, Hey Joe

There are lots of other areas to explore when looking at the relationship between maths and music – one of which is looking at how we can model the wave frequencies of notes using modified sine/cosine curves.  The IB have included a piece of coursework on this as an example for the new exploration topics.

Another interesting exploration is looking at the strange properties of the Harmonic Sequence – which is the sequence 1, 1/2, 1/3, 1/4… This sequence like many of those found in music is said to be in harmonic progression .  There are some interesting paradoxes related to the harmonic sequence – and a variety of methods of proving that the sum of this sequence (the series) actually diverges to infinity – even though you would intuitively expect it to converge.  The video below provides a taster on this topic:

If you liked this post you might also like:

Synesthesia – Do Your Numbers Have Colour? What happens when 2 senses get cross-wired in the brain.

Wau: The Most Amazing Number in the World? A great video by Vi Hart – see if you can spot the twist!

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

A Maths Snooker Puzzle

This was suggested by Paul our Physics teacher – and is a nice little maths puzzle.

The maximum break score in snooker is 147 which is achieved by:
15 reds (1 point each) , 15 blacks (7 points each), then yellow ( 2 points), then green (3 points), brown (4 points), blue (5 points), pink (6 points) and finally black (7 points).

Now, if you wanted the maximum break in snooker to be 180, but wanted all the balls to still have a distinct value and for black to remain the highest ball, how could you change the values of the balls to get a 180 top score? And can you prove there is only one answer?

Answer below in white text (highlight to reveal)

1) Firstly we can show that if red has to be 1.  For red-black combinations (2,8) (2,9) we can show that no solution is possible.  For (2,10) and above, and for (3,9) and above we reach 180 without the other colours.
2) The only options are (1,8) (1,9) or (1,10).  (1,8) leaves us needing 37 from 5 balls valued between 2-7 – which we can’t do.  (1,10) leaves us needing 5 from 5 balls – which we can’t do.  (1,9) is the only possible solution – and this requires 21 from 5 balls valued (2,3,4,5,6,7,8).  This can only be achieved with 2,3,4,5,7.

Like puzzles?  Then you might also enjoy some other brain teaser posts here.

Is God a Mathematician?

That’s the provocative question posed by American Physicist Michio Kaku in this fascinating 5 minute interview which takes in the ideas of Newton, Einstein and modern ideas on String Theory.  It addresses the fundamental questions in maths ToK – is mathematics invented or discovered?  What explains the “unreasonable effectiveness” of mathematics in the universe?

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

Mathematician Eugene Wigner (quoted above) has called this seemingly fundamental relationship between mathematics and our world as nothing short of miraculous.  Why should the motion of planets, the interaction of subatomic particles, the energy of a Black Hole, the growth of populations, the energy in a spring all  be described by mathematical equations?

Since the invention of quantum mechanics in the 1920s, mathematicians have been searching for a grand unified theory – a so called Theory of Everything – which can describe the entire universe through a single mathematical equation.  We have quantum mechanics which describes the sub-atomic world with remarkable accuracy, and general relativity which does equally well with describing the macroscopic world.  However the 2 theories break down when objects are both massive and subatomic (such as the singularity in a Black Hole or the conditions at the birth of the Universe) – hence the need for a single unified theory that can bridge this gap.

Another Michio Kaku video talking through the problem with describing the singularity at the heart of a Black Hole

For a more in-depth look at this topic, read, “Is Maths Invented or Discovered?”

This blog was inspired by a video posted on Larry Ferlazzo’s ToK site – which has hundreds of links and ideas for ToK maths articles in the news – well worth reading!

This is a great resource from Mr Collins – Maths Pictionary. What I like about this is that it can be incorporated into a large number of classroom activities – from Jeopardy games, to starters to topic revision.  It can also be easily adapted to everything from KS3 to IB – and can be a great way of revising key vocabulary.

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### IB Maths Super Exploration Guide

A Super Exploration Guide with 168 pages of essential advice from a current IB examiner to ensure you get great marks on your coursework.