The Gambler’s Fallacy and Casino Maths

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.

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

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Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

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.