Introduction

What is probability?

• Give an example and ask what it means that an event has probability `p'.

• Frequentist definition of probability. An event has a probability $p$ if the long run relative frequency of the event approaches $p$ when a large number of identical, independent trial is repeated. The problem with this definition of probability is that in many cases, it is impossible or impractical to examine a large number of independent idential trials. For example, how would you determine the probability of a nuclear reactor failing?

• Model definition of Probability. In some cases, probabilities can be derived by developing a model - a theoretical constuct that approximates reality. For example, in models where all outcomes are equally likely, the probability is simply defined as a 1/number of outcomes. For example, in flipping a coin, one model may postulate that both sides are equally likely to occur; hence the probability of a head is defined a 1/2. In determining the sex of an unborn child, the model may postulate that both genders are equally likely and hence the probability of a girl is defined a 1/2. The real danger is that your model may be wrong - for example, in actual fact, the probability of a girl in a birth is only .48.

• Subjective probability In some cases, probabilities are assigned on the basis of prior belief in certain outcomes. For example, what is the probability that the you will pass this course? In this case, the concept of a large number of identical trials is meaningles; the idea of a model is silly. The number that you select is based on your subjective knowledge of your ability and how much you plan to study. Clearly the problem with subjective probabilities is that two people may assign quite different probabilities to the same event and there is no objective way of defending either position. Subjective probabilities are quite common - for example whenever someone declares that the chances of something happening are 1 in a million, they are using a subjective assessment of probabiity.

The law of large numbers and the lawless of small numbers.

The Law of Large Numbers simply states that in a large number of trials, the observed frequencey of events will be close to the theoretical probability. For example, in a large number of flips of a fair coin, the proportion of heads should be close to .50.

The Lawless of Small Numbers simply states that in the above law, a ``large number'' is LARGE!

These two concepts are intuively understood by many people, but they fail to understand just how large is LARGE. For example, in the gamblers fallacy, the gambler will observe that in the last 10 flips of a coin, 8 heads occured. The person will argue ``Therefore in order for the probabilities to even out, the chances of a head must be less on the next few flips.'' Unfortunately, while it is true that the number of heads will be less than .50 in remainin flips, this is over millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions, and millions of next flips (you get the idea!) The Law of Large Numbers simply states that in the LONG RUN, and the LONG RUN is very long indeed, the observed occurances will come close to the true probabilities, but say nothing about what can happen in the short run.

In general, sample statistics give information about data already collected while probability has information about future events from the entire population. Remember the alliteration `Statistics are for Samples; Probability is for Populations'.