MATHEMATICS 2
M2.4: Probability
M2.4.A The Fundamental Counting Principle
Introduction
The Fundamental Counting Principle is the guiding rule for finding the number of ways to accomplish two tasks, or to discover the number of combinations of two events.
If there are m ways to do one thing, and n ways to do another, then there are m * n ways of doing both. To put it another way, if one event occurs m ways and another occurs n ways, then the number of combinations of events that can occur is m times n.
These videos discuss the Fundamental Counting Principle. The classes and textbooks from which they were developed is not ours, so please ignore the course names and chapter numbers. If you have any difficulties with the videos, you may want to check the plugins for your browser.
If there are m ways to do one thing, and n ways to do another, then there are m * n ways of doing both. To put it another way, if one event occurs m ways and another occurs n ways, then the number of combinations of events that can occur is m times n.
These videos discuss the Fundamental Counting Principle. The classes and textbooks from which they were developed is not ours, so please ignore the course names and chapter numbers. If you have any difficulties with the videos, you may want to check the plugins for your browser.


Fundamental Counting Principles and Basic Probability
An event in a discrete sample space S is a collection of sample points, i.e., any subset of S. In other words, an event is a set consisting of possible outcomes of the experiment.
A simple event is an event that cannot be decomposed. Each simple event corresponds to one and only one sample point. Any event that can be decomposed into more than one simple event is called a compound event.
A classical probability is the relative frequency of each event in the sample space when each event is equally likely. The formula for classical probability of an event occurring is the number in the event divided by the number in the sample space. This is only true when the events are equally likely. It is represented as follows:
A simple event is an event that cannot be decomposed. Each simple event corresponds to one and only one sample point. Any event that can be decomposed into more than one simple event is called a compound event.
A classical probability is the relative frequency of each event in the sample space when each event is equally likely. The formula for classical probability of an event occurring is the number in the event divided by the number in the sample space. This is only true when the events are equally likely. It is represented as follows:
P(E) = n(E) / n(S)
Probability Rules
All probabilities are between 0 and 1 inclusive. This is represented as follows:
0 <= P(E) <= 1
To put it another way, the sum of all the probabilities in the sample space is 1.
There are some other rules which are also important.
The probability of an event which cannot occur is 0.
The probability of any event which is not in the sample space is zero.
The probability of an event which must occur is 1.
The probability of the sample space is 1.
The probability of an event which cannot occur is 0.
The probability of any event which is not in the sample space is zero.
The probability of an event which must occur is 1.
The probability of the sample space is 1.
The probability of an event not occurring is one minus the probability of it occurring. This is represented as follows:
P(E') = 1  P(E)
The following videos concentrate on the basic rules of probability. If you have any difficulties playing the videos, you may want to check the plugins for your browser.


More Probability Rules
"OR", or Unions 
"AND", or Intersections

Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint.
If two events are disjoint, then the probability of them both occurring at the same time is 0. This is represented as follows: Disjoint: P(A and B) = 0
If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring.
This is the Specific Addition Rule, and is only valid when the events are mutually exclusive. It is represented as follows: P(A or B) = P(A) + P(B)
The probability of an event not occurring is one minus the probability of it occurring. This is represented as follows:
P(E') = 1  P(E)

Independent Events
Two events are independent if the occurrence of one does not change the probability of the other occurring.
If events are independent, then the probability of them both occurring is the product of the probabilities of each occurring. This is the Specific Multiplication Rule, and is only valid for independent events. It is represented as follows: P(A and B) = P(A) * P(B)

NonMutually Exclusive Events In events which aren't mutually exclusive, there is some overlap. When P(A) and P(B) are added, the probability of the intersection (and) is added twice. To compensate for that double addition, the intersection needs to be subtracted. This is the General Addition Rule, and is always valid. It is represented as follows: P(A or B) = P(A) + P(B)  P(A and B) 
The discussion of dependent events is reserved for Objective M2.4.B, Conditional Probability

Exploration of Knowledge and Tools
When you have finished viewing these videos, please participate in the poll. When the number of votes indicates that everyone has reached this point, I will unlock the Probability Exploration Room so we can explore the topic as a group.
(Due to the nature of course design, for the moment the room is unlocked; please go to the Probability Exploration Room when you are done with the poll). When you return, please post the answers to your questions in the Probability Blog. When you post your answers, I will email the homework assignment to you. 

Reflection and Revision
You will receive your homework assignment when you complete the exploration activities. Complete the following homework assignments individually by the due date listed on the Course Schedule. You can create Word documents, .txt files, .rtf files, .gifs, .jpgs, or .zip files that contain those formats; all other document formats will be rejected. Attach your email to the homework assignment and return it to me at [email protected].
When I receive your homework, I will assign you to groups of 3 for the next exercise. The order in which I receive the assignments (i.e., the timestamp from the Internet Service Provider) will determine the members of the groups.
When I receive your homework, I will assign you to groups of 3 for the next exercise. The order in which I receive the assignments (i.e., the timestamp from the Internet Service Provider) will determine the members of the groups.
Assessment and Observation
Group Collaboration
For this competency goal, you have learned about the Fundamental Counting Principle, probability rules and their relationships to each other. You have constructed probabilities algebraically and on your calculators. You have described equations using new terminology such as “specific multiplication rule” and “nonmutually exclusive event”.
Observations verify information transfer and knowledge construction because you have had several opportunities to interact with the lesson material and reinforced your knowledge with individual homework. The following activity will assess your competency in groups of 3 to solve problems. You have a week to collaborate.
Group 1: Student, Student, Student (your name here)
Group 2: Student, Student, Student (your name here)
Group 3: Student, Student, Student (your name here)
Group 4: Student, Student, Student (your name here)
Group 5: Student, Student, Student (your name here)
Group 6: Student, Student, Student (your name here)
Group 7: Student, Student, Student (your name here)
Observations verify information transfer and knowledge construction because you have had several opportunities to interact with the lesson material and reinforced your knowledge with individual homework. The following activity will assess your competency in groups of 3 to solve problems. You have a week to collaborate.
Group 1: Student, Student, Student (your name here)
Group 2: Student, Student, Student (your name here)
Group 3: Student, Student, Student (your name here)
Group 4: Student, Student, Student (your name here)
Group 5: Student, Student, Student (your name here)
Group 6: Student, Student, Student (your name here)
Group 7: Student, Student, Student (your name here)
If you have questions or comments, please contact me at [email protected], leave comments on the Guest Blog, or use the form on the Contact Us page.