The Gambler’s Fallacy

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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.

The house (i.e., the casino) takes a set percentage out of the payout of every possible bet when you win a bet (except the Free Odds bet, which is discussed in another article). The easiest way to illustrate this concept is to compare the results of making a Place bet on the number 4 or 10. (We’ll look at Place bets in another article.). The “theo” or “theoretical” is the actual dollar amount the casino expects to win from the player. The “return to player” measures the fraction of the player’s total wagers the player is paid back in live casino play. X-FOR-1, X-TO-1, 1-IN-X Most table game wagers are paid in an 'X-to-1' fashion.

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.

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.

Reading time: ~15 min

Soon after their initial discovery, mathematicians started applying the laws of probability to many different parts of life – including casino games.

One of these mathematicians was Karl Pearson who analysed the results of roulette games published in the French newspaper Le Monaco.

Roulette consists of a wheel with the numbers from 1 to 36 coloured in red and black, as well as a green 0. A ball rolls around the outside and randomly lands on one of the numbers. Gamblers can bet on a single number, a set of multiple numbers, or just a colour. Their potential winning depends on the likelihood of each of these outcomes.

Here is one of the many hundreds of newspaper extracts which Pearson collected and analysed. At first sight, it looks pretty random:

Roulette results on 19 August 1823, Table 5:

1312303331229582223135181431361518283229113423368162392016141526312115333221214963013335281727653411183269312921835613428110

A roulette wheel has the same number of red and black numbers. If we ignore the green 0 (which means the casino wins) we would expect the number of red and black numbers to be .Let’s check that this is indeed the case for the set of results above.

Black

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This looks pretty evenly distributed – there is a small difference between the number of red and black results, but that is always to be expected in probability.

However, Pearson didn’t stop here. He realised that if the results were completely random, then each of the four possible pairs of two consecutive colours should also be equally likely. Again we can count the number of occurrences in our example:

RB

BB

For some reasons, it seems that RR and BB happen much than RB and BR, even though they should all have the same probability. Of course, we might have just been unlucky in this particular sequence of results – but Pearson tested many thousands of results and always found the same.

It gets even wo

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