In this chapter we consider the exact meaning of our everyday word “chance.”  We talk about chance in various ways: “there is a good chance of rain today,”  “they have no chance of winning,” “there is about one chance in three,” and so on. In some cases the meaning is very vague, but sometimes there is a precise numerical meaning. We shall use the word “probability” to formalize those cases where “chance” has a precise meaning, and we shall assign a numerical value to probability

Some Definitions

One way to think about the probability that an event will happen: suppose the same circumstances were to occur a great many times. In what fraction of cases would the event occur? This fraction is the probability that the event occurs. So probabilities will lie between 0 and 1; 0 represents impossibility, 1 represents absolute certainty. Often people express probabilities as percentages, rather than fractions. For example, consider the question: What is the chance it will rain tomorrow? We could ask, if the exact circumstances (current weather, time of year,  worldwide wind patterns, and so on) were reproduced in a million cases, in what fraction would it rain the next day? And while we cannot actually make these circumstances occur, in practice we can try to get a good estimate using weather records and geographical/geophysical theory.

Many problems will involve ordinary dice, as used for example in games like Monopoly. These have six faces, with the numbers 1–6 on them. Dice can be biased,  so that one face is more likely to show than another. If we roll an ordinary, unbiased die, what is the probability of rolling a 5? The six possibilities are equally likely, so the answer is 1/6 . If the die were biased, you might try rolling a few hundred times and keeping records.

Another idea that is commonly used in probability problems is the deck of playing cards. A standard deck has 52 cards; the cards are divided into four suits:

Diamonds and Hearts are red, Clubs and Spades are black. Each suit contains an Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3 and 2, and these are called the 13 denominations. (The Ace also doubles as a 1.) When a problem says “a card is dealt from a standard deck” it is assumed that all possible cards are equally likely, so the probability of any given card being dealt is 1/52 . There are four cards of each denomination, so the answer to “what is the probability of dealing a Queen,” or any other fixed denomination, is 4/52 , or 1/13 .

We say an event is random if you can’t predict its outcome for sure. This does not mean the chances of different outcomes are equal, although sometimes people use the word that way in everyday English. For example, if a die is painted black on 5 sides, white on one, then the chance of black is 5/6 and the chance of white is 1/6 .

This is random, although the two probabilities are not equal.

When we talk about the outcomes of a random phenomenon, we mean the distinct possible results; in other words, at most one of them can occur, and one must occur.

The set of all possible outcomes is called the sample space. Each different outcome will have a probability. These probabilities follow the following rules:

1. One and only one of the outcomes will occur.

2. Outcome X has a probability, P(X), and 0 ≤ P(X) ≤ 1.

3. The sum of the P(X), for all outcomes X, is 1.