Alex's Adventures in Numberland - Alex Bellos [126]
Superstition presented a powerful block against a scientific approach to probability, but after millennia of dice-throwing, mysticism was overcome by perhaps a stronger human urge – the desire for financial profit. Girolamo Cardano was the first man to take Fortune hostage. It could be argued, in fact, that the invention of probability was the root cause of the decline, over the last few centuries, of superstition and religion. If unpredictable events obey mathematical laws, there is no need to have them explained by deities. The secularization of the world is usually associated with thinkers such as Charles Darwin and Friedrich Nietzsche, yet quite possibly the man who set the ball rolling was Girolamo Cardano.
Games of chance have most commonly involved dice. A model popular in antiquity was the astragalus – an anklebone from a sheep or a goat – that had four distinctly flat faces. Indians liked dice in the shape of rods and Toblerones, and they marked the different faces with pips, the most likely explanation for which is that dice predate any formal system of numerical notation, and this tradition has survived. The fairest dice have identical sides, and if one imposes the further condition that each side must also be a regular polygon, there are only five shapes that fit the bill, the Platonic solids. All the Platonic solids have been used as dice. Ur, possibly the world’s oldest known game, which dates from at least as far back as the third century bc, used a tetrahedron, which, however, is the worst of the five choices because the tetrahedron hardly rolls and has only four sides. Octahedrons (eight sides) were used in ancient Egypt, and dodecahedrons (twelve) and icosahedrons (twenty) are still found in fortune-tellers’ handbags.
By far the most popular dice shape has been the cube. It is the easiest to make, its span of digits is neither too big nor too small, it rolls nicely but not too easily, and it’s obvious on which number it lands. Cubical dice with pips are a cross-cultural symbol of luck and chance, as comfortable bashed around in Chinese mah-jong salons as they are dangling from the rear-view mirrors of British cars.
As I mentioned earlier, roll one die and the chance of a six is . Roll another die and the chance of a six is also . What is the chance of rolling a pair of dice and getting a pair of sixes? The most basic rule of probability is that the chances of two independent events happening is the same thing as the chance of one event happening multiplied by the chance of the second event happening. When you throw a pair of dice the outcome of the first die is independent of the outcome of the second, and vice versa. So, the chances of throwing two sixes is × , or . You can see this visuallgits iounting all the possible combinations of two dice: there are 36 equally likely outcomes and only one of them is a six and a six.
Conversely, of the 36 possible outcomes, 35 of them are not double six. So, the probability of not throwing a six and a six is . Rather than counting up 35 examples, you can equally start with the full set and then subtract the instances of double sixes. In this example, 1 – =. The probability of something not happening, therefore, is 1 minus the probability of that thing happening.
The dicing table was an early equivalent of the slot machine, where gamblers placed wagers on the outcome of dice throws. One classic gamble was to roll four dice and bet on the chance of at least one six appearing. This was a nice little earner for anyone willing to put money on it, and we already have enough maths knowledge to see why:
Step 1: The probability of rolling a six in four rolls of dice is the same as 1 minus the probability of not getting a six on any of the four dice.
Step 2: The probability of not getting a six on one die is , so if there are four dice, the probability