1 - A SINGLE RANDOM VARIABLE Questions involving probability abound in Computer Science: · What is the probability of the PWF world falling over next week? · What is the probability of one packet colliding with another in a network? · What is the probability of an undergraduate not turning up for a lecture? When addressing such questions there are often added complications: the question may be ill posed or the answer may vary with time. Which undergraduate? What lecture? Is the probability of turning up different on Saturdays? Let's start with something which appears easy to reason about ... Introduction - Throwing a die Consider an experiment or trial which consists of throwing a mathematically ideal die. Such a die is often called a fair die or an unbiased die. Common sense suggests that: · The outcome of a single throw cannot be predicted. · The outcome will necessarily be a random integer in the range 1 to 6. · The six possible outcomes are equiprobable, each having a probability of 7. Without further qualification, serious probabilists would regard this collection of assertions, especially the second, as almost meaningless. Just what is a random integer? Giving proper mathematical rigour to the foundations of probability theory is quite a taxing task. To illustrate the difficulty, consider probability in a frequency sense. Thus a probability of 1 means that, over a long run, one expects to throw a 5 (say) on one-sixth of the occasions that the die is thrown. If the actual proportion of 5s after n throws is p5(n) it would be nice to say: n->00 lim ps (n) = 1 6 Unfortunately this is utterly bogus mathematics! This is simply not a proper use of the idea of a limit. Suppose you frequently play a board game which requires you to 'throw a 6 to start'. Each time you play the game you write down how many throws it took to get your 6. One day you ponder what the long-term average ought to be .. . You reason that each of the six possible outcomes should occur with equal frequency in each of the first six throws. You reason further that the required 6 could be in any position from first to sixth in these first six throws and so, on average, you will get the first 6 at throw 32. This, too, is nonsense reasoning but you will have to wait a while before the correct answer is derived. Meantime, note a health warning: relying on intuition can seriously damage your understanding of Probability! - 1.1 -
Relationship to Set Theory - Sample Space - Sample Point This course will concentrate on solving problems. It is not proposed to venture very far into the foundations of probability theory but some formal discussion is unavoidable. It is mathematically sound to represent the collection of possible outcomes of an experiment as a set. Call this set 22. In the case of a die: 2 = {1, 2, 3,