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:
Although the game of roulette has grown in popularity, there are still many misconceptions about it. In this article, we highlight 4 roulette myths which are often told by gamblers – but which in reality aren’t true at all. Let’s start debunking these myths one by one! Myth 1: There is only one version of roulette. Gambler’s Fallacy The gambler’s fallacy is the erroneous belief that if an independent event happens frequently (or infrequently), it impacts the chances of it happening again. So, if the roulette ball landed on red three times in a row, it’s more likely to land on black in the next spin, right?
Russian Roulette Fallacy
$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.
Roulette Strategy Gambler's Fallacy
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.
Gamblers Fallacy (Monte Carlo Fallacy) is the false belief that past events have an impact on the probability of future outcomes of a truly random event.
Gamblers Fallacy Example
Roulette Fallacy
Contents
The easiest way to explain this is with an example. If we flip a “fair coin” (a truly balanced coin with 50/50 chance of being heads or tails) ten times and it lands on heads 9 times, someone who believes the gambler’s fallacy will wrongly assume that the next flip is “due” to be tails (tails is overdue as an outcome).
In reality this is wrong. The rules of probability say that every “fair coin” flip has exactly a 50/50 chance of landing on heads or tails no matter what. Period, end of story. This applies very easily to roulette, it doesn’t matter what color came last on a fair roulette table. The probability of hitting red or black doesn’t change depending on passed outcomes either.
Russian Roulette Fallacy
MYTH: Despite what some people think, Roulette doesn’t have 50/50 odds, there is 1 green zero in European Roulette and 2 green zeros in American Roulette.
Inverse Gamblers Fallacy
Unsurprisingly inverse Gamblers Fallacy (or reverse Gamblers Fallacy) is the mistaken belief that the coin flipped 9 times on heads will land on heads. This is also not true for the same reasons as above, it is however a better bet. This is because in reality nothing is perfectly designed. If a coin landed on heads 9 times in a row, one might assume the coin favored heads in design (it wasn’t a fair coin, thus one should bet heads to materialize an edge inherent in this unfair coin).
FACT: Future odds are not based on past odds. The probability of something happening more than once, before any event takes place, is not the same as the probability of each event happening. The chances of a coin landing on heads ten times is 1/2^10 or 1 in 1032, if a coin has already landed on heads 9 times the chances of it land on heads again is 1/2^1 or 1 in 2.
Roulette Fallacy
The Psychology of Gamblers Fallacy
Without getting too deep into things, we are hardwired to recognize patterns. Your brain wants to tell you that outcomes can be predicted based on past events (fire burns hand, brain says don’t touch fire). However, for the purposes of gambling one should not trust their instincts and always trust a solid strategy based on mathematics if their intention is to actually win.