To the layman an event is something that happens. To the statistician, an event is an outcome or set of outcomes of an experiment. This brings up the question of what is an experiment. For our purposes an experiment will be any activity that generates an observable outcome. Some simple examples of experiments include flipping a coin, rolling a pair of dice, and drawing cards from a deck. An outcome of each of these experiments could be “heads”, 7, or the ace of hearts respectively. For the example of the coin flip or drawing a card from the deck, the example outcomes given can be thought of as “atomic” in the sense that they cannot be further broken down into simpler events. The outcome of achieving a 7 on a roll of a pair of dice could be thought of as consisting of a pair of outcomes, one for each die. For example the 7 could be the result of 2 on the first die and 5 on the second. Having a flipped coin land heads up cannot be similarly decomposed. An event can also be thought of as a collection of outcomes rather than just a single outcome. For example the experiment of drawing a card from a standard deck, the events could be segregated into hearts, diamonds, spades, or clubs depending on the suit of the card drawn. Then any of the atomic events 2, 3, . . . , 10, jack, queen, king, or ace of hearts would be a “heart” event for the experiment of drawing a card and observing its suit.
In this chapter the outcomes of experiments will be thought of as discrete in the sense that the outcomes will be from a set whose members are isolated from each other by gaps. The discreteness of a coin flip, a roll of a pair of dice, and card draw are apparent due to the condition that there is no outcome between “heads” and “tails”, or between 6 and 7, or between the two of clubs and the three of clubs respectively. Also in this chapter the number of different outcomes of an experiment will be either finite or countable (meaning that the outcomes can be put into one-to-one correspondence with a subset of the natural numbers).
The probability of an event is a real number measuring the likelihood of that event occurring as the outcome of an experiment. To begin the more formal study of events and probabilities, let the symbol A represent an event. The probability of event A will be denoted P (A). By convention, probabilities are always real numbers in the interval [0, 1], that is, 0 ≤ P (A) ≤ 1. If A is an event for which P (A) = 0, then A is said to be an impossible event. If P (A) = 1, then A is said to be a certain event. Impossible events never occur, while certain events always occur. Events with probabilities closer to 1 are more likely to occur than events whose probabilities are closer to 0.
There are two approaches to assigning a probability to an event, the classical approach and the empirical approach. Adopting the empirical approach requires an investigator to conduct (or at least simulate) the experiment N times (where N is usually taken to be as large as practical).
During the N repetitions of the experiment the investigator counts the number of times that event A occurred. Suppose this number is x. Then the probability of event A is estimated to be P (A) = x/N. The classical approach is a more theoretical exercise. The investigator must consider the experiment carefully and determine the total number of different outcomes of the experiment (call this number M), assume that each outcome is equally likely, and then determine the number of outcomes among the total in which event A occurs (suppose this number is y). The probability of event A is then assigned the value P (A) = y/M. In practice the two methods closely agree, especially when N is very large.
Some experiments involve events which can be thought of as the result of two or more outcomes occurring simultaneously. For example, suppose a red coin and a green coin will be flipped. One compound outcome of the experiment is the red coin lands on “heads” and the green coin lands on “heads” also. The next section contains some simple rules for handling the probabilities of these compound events.
| Example
Suppose the four sides of a regular tetrahedron are labelled 1 through 4. If the tetrahedron is rolled like a die, what is the probability of it landing on 3? Assuming the regular tetrahedron is fair so that it equally likely to land on any of its four faces, the probability of it landing on 3 is p = 1/4. |
| Exercises
1. Use the classical approach and the assumption of fair dice to find the probabilities of the outcomes obtained by rolling a pair of dice and summing the dots shown on the upward faces. 2. Part of a well-known puzzle involves three people entering a room. As each person enters, at random either a red or a blue hat is placed on the person’s head. The probability that an individual receives a red hat is 1/2. No person can see the colour of their own hat, but they can see the colour of the other two persons’ hats. The three will split a prize if at least one person guesses the colour of their own hat correctly and no one guesses incorrectly. A person may decide to pass rather than to guess. The three people are not allowed to confer with one another once the hats have been placed on their heads, but they are allowed to agree on a strategy prior to entering the room. At the risk of spoiling the puzzle, one strategy the players may follow instructs a player to pass if they see the other two persons wearing mis-matched hats and to guess the opposite colour if their friends are wearing matching hats. Why is this a good strategy and what is the probability of winning the game?
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