frequency viewpoint will still be useful in order to gain intuitive understanding. And by the end of the book, we will actually derive the relative frequency viewpoint as a consequence of the mathematical theory.
1.3 PROBABILITY FUNCTION
We assume for the next several chapters that the sample space is discrete. This means that the sample space is either finite or countably infinite.
A set is countably infinite if the elements of the set can be arranged as a sequence. The natural numbers
If the sample space is finite, it can be written as
The set of all real numbers is an infinite set that is not countably infinite. It is called uncountable. An interval of real numbers, such as (0,1), the numbers between 0 and 1, is also uncountable. Probability on uncountable spaces will require differential and integral calculus and will be discussed in the second half of this book.
A probability function assigns numbers between 0 and 1 to events according to three defining properties.
PROBABILITY FUNCTION
Given a random experiment with discrete sample space
1
2 (1.1)
3 For all events ,(1.2)
You may not be familiar with some of the notation in this definition. The symbol
In the case of a finite sample space
And in the case of a countably infinite sample space
In simple language, probabilities sum to 1. The third defining property of a probability function says that the probability of an event is the sum of the probabilities of all the outcomes contained in that event. We might describe a probability function with a table, function, graph, or qualitative description. Multiple representations are possible, as shown in the next example.
Example 1.5 A type of candy comes in red, yellow, orange, green, and purple colors. Choose a piece of candy at random. What color is it? The sample space is Assuming the candy colors are equally likely outcomes, here are three equivalent ways of describing the probability function:
1 0.200.200.200.200.20
2
3 The five colors are equally likely.
In the discrete setting, we will often use probability model and probability distribution interchangeably with probability function. In all cases, to specify a probability function requires identifying (i) the outcomes of the sample space and (ii) the probabilities associated with those outcomes.
Letting H denote heads and T denote tails, an obvious model for a simple coin toss is
Actually, there is some extremely small, but nonzero, probability that a coin will land on its side. So perhaps a better model would be
Ignoring the possibility of the coin landing on its side, a more general model is
where
