Learning Resources

Lesson

In the first lesson in this unit we learned the formula for calculating theoretical probability. It stated that the probability a given event X will occur, denoted as P(X), is calculated as follows:

                    

To use this formula, we have to be able to count the numbers in both the numerator and denominator. As we have said before,  theoretical probability involves counting.  In the examples discussed so far, the counting has been fairly easy. However, there are situations that present some difficulties when we try to count them.

The rest of this unit is devoted to devising methods of counting that will assist us in calculating theoretical probability. The first of these methods is referred to as the Fundamental Counting Principal. To help understand it consider the following example.

Example

If a coin is tossed and a six sided die is rolled, what is the probability we will get a head on the coin and a 5 on the die?

Solution

We have to count the number of ways the event can happen and compare it to the total number of possible outcomes. One way of doing this is to construct a tree diagram, which those of you who have done Math 2204 should remember how to do. Use the demo below to generate the tree diagram for this problem. You can generate it in a step by step fashion of view the completed tree.

From the above "tree" you should be able to see that the total number of possible outcomes for this situation is 12 and the number of ways the desired event could occur is 1. Thus we can say:

Note that there were two possibilities for the coin (head or tail) and six possibilities for the die (1, 2, 3, 4, 5, 6). The aim of this investigation is to see how to use these numbers to calculate the total number of possible outcomes without having to draw a tree diagram.

Complete Investigation 3 on page 307 in your text. Then go to the top of the page and click on the Activities button. If you are still not sure what the Fundamental Counting principle states continue reading below.

The Fundamental Counting Principle states that if an event consists of several parts or stages, then the size of the sample space is equal to the product of the numbers of possibilities that exist for each part or stage.

Example

A television manufacturer has 8 different sizes to choose from. Each size can be purchased with or without the option of a built in VCR, a built in DVD player, or both. The TVs come in black, ivory and metal casings. How many different "models" do they produce?

Solution

                     8 (sizes) X 4 (options) X 3 (casings) = 96 "models"

Activity

1. If you have not already done so, complete Investigation 3 on page 307 in your text.
2. Complete the Investigation Questions 1 & 2 on page 308.
3. Do the CYU Questions 3 - 7 on pages 308 & 309.
4. Read through the example in Focus B on pages 309 & 310.
5. Do the CYU Questions 10 - 12 on page 310.

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

For lunch, a restaurant offers a full meal deal which has the following options: a salad or bowl of soup; a sandwich of turkey, chicken, roast beef, or salami; a drink of coffee, tea, or soft drink; a dessert of apple pie or fruit cup. Just as you finish your meal you notice two friends in a corner booth. When you spoke to them you found out they each had a different option for lunch. What is the probability that you had exactly the same option as one of them?

Solution