Elementary Probability

Part 2: Terminology

In order to discuss the notion of probability in more detail, we need to introduce some terminology. First, note that the word random is often used to describe an activity where the outcome is uncertain. In the urn example of Part 1, we would say that we are selecting a ball "at random" because we do not know the outcome of this activity.

We will discuss the probability of various outcomes in the context of a well-defined activity or procedure. We use the term experiment for such a well-defined procedure. An example is the procedure of selecting a single ball out of the urn. When an experiment is performed, the result is called an outcome. In our urn experiment, the possible outcomes are A green ball is selected and A red ball is selected. We also need a term for the set of all possible outcomes. We will call this set the sample space. So in our urn example, the sample space is a set consisting of the two outcomes.

Example: Rolling a die. Consider the experiment of rolling a single die. There are six possible outcomes: one of the numbers 1, 2, 3, 4, 5, and 6. So, in this case, the sample space has 6 elements. Assuming that we have a fair die (that is, each side is equally likely to turn up), we assign the probability of each outcome to be the ratio 1/6.

Example: Using a game spinner. Consider the experiment of spinning the pointer on the game spinner pictured below. There are three possible outcomes, that is, when the pointer stops it must point to one of the three colors. (We rule out the possibility of landing on the border between two colors.) Since the red region covers half the area of the spinner, we say that the probability of it pointing to the red area is 1/2. Similarly the probabilites of it pointing to the blue and green areas are 1/3 and 1/6 respectively.

1. Describe the experiment and the sample space for both the die example and the game spinner example.

Events

Not only do we want to assign probabilities to individual outcomes, but also we want to assign probabilities to sets of outcomes, i.e., to subsets of the sample space. A subset of the sample space will be called an event. So, how should we assign probabilities to events?

2. Die Example (continued). Consider again the experiment of rolling a fair die. Here, an event may be identified with a subset of the set of 6 integers {1, 2, 3, 4, 5, 6}. For example, if A is the event the die will show an even number, then A = {2, 4, 6}, what probability would you assign to this event?

3. Game Spinner Example (continued). In the experiment with the fair spinner, let A be the event that the outcome is either red or blue. What is the probability that you would assign to this event?

4. Check to see that this definition agrees with your assignments in Examples 1 and 2.

5. Find the probability of each outcome in a sample space of size n, assuming that each outcome in the sample space is equally likely to occur.

6. Suppose we roll a fair die and A is the event that the outcome shows a number less than 3. Find P(B).

7. Now suppose we roll two fair dice. We can generate the elements of this sample space as ordered pairs. For example, the ordered pair (2, 5), indicates that the first die showed a 2 and the second a 5. Explain why the size of this sample space is 36.

8. In the experiment of rolling a pair of fair dice, let A be the event that the sum of the two faces showing is 5. Find P(A). Clearly indicate how you arrived at your answer. What other sum has the same probability of appearing as the sum 5?

The following applet will allow you to experiment with rolling a pair of dice. Close the applet window when you are done.

9. Did your applet experiments agree with your answer to Question 8