Probability
Instructional Notes
Math B30
Probability
Instructional Notes
Some students may have been introduced to this concept previously (See Mathematics A 30 - concept A. 1). This concept is needed before beginning the section on probability.
Students can be introduced to the concept of inclusion and exclusion by dealing with two sets of objects at the outset, and then expanding this to three or more sets of objects. Students should work individually, or in pairs, with some initial exercises designed to illustrate the principle of inclusion and exclusion.
Venn diagrams should be used to illustrate these concepts.
When students have become familiar with the concepts, formal notation should be introduced.
The term mutually exclusive should be defined and examples provided by/for the students to indicate their understanding of the term. When this is complete, students may attempt some exercises that involve mutually exclusive events. They should work individually, in pairs or in small groups to determine the solutions to these questions. A brief review of the concept of probability may also be necessary. (PSVS)
Some examples they might work on are as follows:
a) Given two Canadian pennies, find the probability of tossing two heads or two tails.
b) Given a set of cards numbered from 1 to 6, find the probability of drawing a 5 or a 6.
Exercises should become progressively more difficult, as the students' understanding deepens. Students should be able to determine the pattern for calculating the number of outcomes for mutually exclusive events, determine if A and B represent mutually exclusive events, then note that P(A or B) = P(A) + P(B).
It is important to note that for any two events that are not mutually exclusive, we have P(A _ B) ¹ 0. The teacher may wish to include some questions of this type as well.
For these, the P(A _ B) = P(A) + P(B) - P(A _ B), which can be related to the principle of inclusion and exclusion.
E.g.: What is the probability of rolling a 4 or rolling an even number?
E.g.: What is the probability of drawing a king or a red card?
The students should be given some problem situation to solve using their knowledge of probability. This could be similar to finding the probability of rolling a five on a die and tossing a head with a coin. The students could do this experimentally, list the sample space, and read the result. The introduction of probability trees in some of these instances may help student understanding. After doing several of these types of questions, the students could be guided to the definition of independent events and the formula used to calculate the probability of independent events, that is, 
It is useful to introduce the definition of the complement of the probability of an event A at this point.
Note that: 
The definition of dependent events is crucial to a valid understanding of this concept. In order to help students attain this understanding, it might be useful to discuss real-world dependent events and their connectedness before doing any calculations.
Example. Using a deck of standard playing cards, discuss the effect of drawing two cards without replacing the first card drawn. Point out that it would be impossible to redraw the same card, that there would only be 51 (instead of 52) choices for the second draw, and so on.
Note: Many students are not familiar with playing cards, so a sample pack may have to be provided by the teacher or other students.
Have students work individually, in pairs or small groups on an exercise or two that allows them to work with dependent events. As an example, they might be given a paper bag containing three green and four red markers, and a series of exercises such as:
If we pull out two markers, one at a time, without replacing the first, what is the probability that we obtain two red ones? two green ones? one green and one red? (A probability tree may be utilized here.)
The students should eventually become familiar with the formula associated with dependent events.
Note that 
in the case where A and B are independent.
The students should be relatively familiar with the types of exercises posed in this section, but may encounter difficulty in deciding what type of situation is described in each exercise. To allow for such, some opportunity to practise identifying the types when they appear together should be provided. A series of statements representing the various types of exercises could be distributed. Students should identify, with justification, the concept involved. Once the underlying concept is correctly identified, then the students can be instructed to complete the calculations to find the indicated solution. (In some cases, it might be suggested that setting-up the correct solution is all that is required.)
This topic can be introduced in many ways, as indicated in the objective itself. The students should receive enough practice to become familiar with the coefficients of expansion and how to obtain them. It might be useful to actually expand a binomial of the type (x + y)4 by multiplication, so that students actually see that the coefficients are in fact the same as those found using these other techniques.
The techniques most often utilized are Pascal's Triangle, combinations, or multiplication. (It is intended that students will use combinations by the end of this section.)
The teacher might have students complete the following exercise and relate the coefficients to Pascal's Triangle.
Other activities are suggested in the next column.
When students are familiar with these coefficients, they should be given some exercises enabling them to practise not only finding these coefficients but being able to determine the indicated term as well. The combinatorial approach should be emphasized.
Students should be able to perform the expansion of common binomials using the Binomial Theorem. It is expected that students will be able to expand examples such as (x + y)6, (2x - y)4, and (x² + 2yz)5 for test purposes, and will be able to expand more difficult binomials as part of their classroom exercises.
It is important to remember that while binomial expansion is a concept to be studied, many applications deal with a specific term of the expansion.
Most real-world expansions deal with two events that have approximately equal chances of occurring, such as the result of tossing coins, the appearance of boys or girls in family order, or the opening or closing of an electrical circuit. (TL)
When introducing the Binomial Theorem, it may be useful to have students expand some examples such as (x + y)5 by multiplication and note the coefficients, the total of the coefficients, the number of terms, and the relationships of the exponents of each successive term. This may enable the students to understand the Binomial Theorem more easily.
Most of the problems in this section deal with two events that have an equally likely chance of occurring, where several trials of the events are conducted. Students could work on these types of questions individually, in pairs, or in small groups, to practise the skills they have learned, not only in the last two sections but in the entire unit in the problem-solving context. (PSVS)