from cosmos to chaos the science of unpredictability aug 2006

[...]... not going to be easy I have from time to time been involved in teaching the laws of probability to high school and university students, and even the most mathematically competent often find it very difficult to get the hang of it The difficulty stems not from there being lots of complicated rules to learn, but from the fact that there are so few In the field of probability it is not possible to proceed... patterns when there is random noise? In this book I cut a broad swathe through the physical sciences, including such esoteric topics as thermodynamics, chaos theory, life on other worlds, the Anthropic Principles, and quantum theory Of course there are many excellent books on each of these topics, but I shall look at them from a different perspective: how they involve, or relate to, the concept of probability... idea of what it means to say, for example, that the probability of a fair coin landing 8 From Cosmos to Chaos heads-up is one-half, or that the probability of a fair dice showing a 6 when you roll it is 1/6 Your understanding of these statements (and others like them) is likely to fall in one or other of the following two basic categories Either the probability represents what will happen if you toss the. .. simple game show The contestant is faced with three doors, behind one of which is a prize The other two have nothing behind them The contestant is asked to pick a door and stand in front of it Then the cheesy host is forced to open one of the other two doors, which has nothing behind it The contestant is offered the choice of staying where he is or switching to the one remaining door (not the one he first... he first picked, nor the one the host opened) Whichever door he then chooses is opened to reveal the prize (or lack of it) The 14 From Cosmos to Chaos question is, when offered the choice, should the contestant stay where he is, swap to the other door, or does it not matter? The vast majority of people I have given this puzzle to answer very quickly that it cannot possibly matter whether you swap or not... not go through the details of calculating the probability of the other types of hand, but I have included a table showing their probabilities obtained by dividing the relevant number of possibilities by the total number of hands at the bottom of the middle column I hope you will be able to reproduce my calculations! Type of Hand Number of Possible Hands Probability Straight Flush Four of a Kind Full... possibility In the toss of a coin, there are two possible outcomes of the experiment, so our event may be either of these Event A might be that the coin shows heads’ Event B might be that the coin shows tails’ These are the only two possibilities and they are mutually exclusive (they cannot happen at the same time) These two events are also exhaustive, in that they represent the entire range of possible... replacement, as long as the number of draws is much smaller than the number of balls in the urn It is a bit tricky to calculate the expectation value of the binomial distribution, but the result is not surprising: E(X) ¼ np If you toss a fair coin 10 times the expectation value for the number of heads is 10 times 0.5, which is 5 No surprise there After another bit of maths, the variance of the distribution... cards There are therefore 26 red cards, so the probability is 26 lots of 1/52, or one-half The simplest alternative method is to say there are only two possible colours and each colour applies to the same number of cards The probability therefore must be 1/2 What is the probability of dealing a king? There are 4 kings in the pack and 52 cards in total The probability must be 4/52 ¼ 1/13 Alternatively there... Consider the probability of getting, say, five spades To do this we have to calculate the number of distinct hands that have this composition There are 13 spades in the deck to start with, so there are 13 Â 12 Â 11 Â 10 Â 9 permutations of five spades drawn from the pack, but, because of the possible internal rearrangements, we have to divide again by 5! The result is that there are 1287 possible hands The . partly to do with the origins of the subject in the mathematics of gambling games, not generally 2 From Cosmos to Chaos regarded as appropriate pastimes for people of good character. The eminent. each of these topics, but I shall look at them from a different perspective: how they involve, or relate to, the concept of probability. Some of the topics I discuss require a certain amount of. and unpredictability. I have therefore used this as an excuse for introducing some examples from card games and the like. 6 From Cosmos to Chaos N 2 O The Logic of Uncertainty The theory of probabilities