PROBABILITY

Introduction

Probability is an area of mathematics which we use all the time in daily life – and usually without thinking about it. While many aspects are very intuitive, probability becomes much more difficult when thinking about it in a precise, mathematical way. The easiest definition is that probabilities are numbers between 0 and 1 which specify the chance that something happens. A probability of 0 means that something is impossible; a probability of 1 means that something is certain.

It is certain that the Earth will still exist tomorrow and impossible that you will meet a basilisk. A coin landing heads or a die rolling 6 have probabilities in between 0 and 1.

However, when tossing a coin, it will either land heads or it will land tails. So why isn’t the probability either 1 or 0 – rather than a number inbetween? Or suppose we toss a coin but don’t look at it. For us, the probability of landing heads is still 0.5, even though it has already happened!

These questions can be answered by thinking about the various different ways in which we can interpret probabilities:

FREQUENTIST The probability of landing heads is the proportion of heads we get if we toss a coin many times.

PROPENSITY The probability of landing heads is the ‘tendency’ of a coin to land heads.

SUBJECTIVIST The probability of landing heads tells us how strongly we ‘believe’ that a coin will land heads.

Notice that subjectivist probabilities may be different for different people – often depending on how much they know. For example, I might estimate that the chance of rain is about 70%, while a meterologist with detailed weather data might say the chance of rain is 64.2%.

Going back to the example of tossing coins, a probability 0.5 of landing heads tells us two things: before tossing a coin we know that it is just as likely to land heads as it is to land tails. And after tossing a coin many times we know that approximately half of the results are heads – though we can’t predict which individual tosses landed heads.

Probability theory is about determining the likelihood of certain events, the chance that something particular is going to happen. In statistics you do exactly the opposite: you look at the results of particular observations and try to determine the probabilities from which they might have originated.

The Meaning of Randomness

© Andrew Davidhazy

Suppose we toss a coin: the chance of it landing heads is 0.5. If we knew which way the coin was facing just before it left the hand, we might be able to make a slightly better predictions – such as 0.58 or 0.41. If we also knew the position, velocity, weight and size of the coin as it left the hand, we could use the laws of physics – gravity, friction and air resistance – to model the motion of the coin and to predict the outcome. Finally, if we knew the exact position of every atom in the coin and of all the air molecules surrounding it, we could create a computer simulation to accurately predict what will happen to the coin.

Therefore one could argue that tossing a coin really isn’t random at all – it is chaotic. This means that the underlying physical principles are very complicated and even tiny changes to the initial conditions (e.g. the speed of the coin) could completely change the result. We can use coins in games and gambling simply because it is so incredibly difficult(and for practical purposes, impossible) to calculate which face it will land on.

The same principle applies to many other “random” events in life: rolling dice, spinning a roulette wheel, or even predicting the weather. They are not really random, we simply don’t have the tools to do the mathematical calculations accurately enough to predict the outcome.There are many other clever way to create random numbers, some relying only on mathematics, others on physical observations such as random variations in atmospheric noise. Yet, it seems that if we had sufficiently accurate data about the physical conditions, we could predict the outcome.On computers you can generate random numbers. These random numbers have many uses, from statistical modelling in Excel to determining who wins a battle in a video game. But, like before, these numbers aren’t really random. For example, a computer could use the last digit of the number of milliseconds since the last startup. If the time since the last startup is, say, 397542 milliseconds, we choose the random number 2. This method is not truly random, but it is so unpredictable that it is random enough for most purposes.