Fundamental Counting Principle:
Suppose two operations are to be performed in order:
With possible outcomes for the first, and for each of these there are possible outcomes for the second. Hence, the total number of possible outcomes is given by the product .
A permutation is an ordered arrangement of a set of objects in a row.
For any natural number , we define .
Suppose r objects are selected from a set of n objects without regard to order, each such selection is called a combination, denoted by or .
Theorem of Combination:
The number of combinations taken r at a time of a set of n objects is given by
The probability of any event H is the sum of the probabilities of those outcomes of the sample space which belongs to denoted by
Probability of Successes for H:
Theorem I of Overlapping Events: or and
Theorem II of Disjoint Events:
If and are mutually exclusive events, then or .
Theorem of Dependent Events:
If the probability of an event A depends on the occurrence of an event B, then
and where the probability that if has occurs,
Theorem of Independent Events:
If A and B are independent events then and .
Rapid Study Kit for "Title":
Core Concept Tutorial
Problem Solving Drill
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"Title" Tutorial Summary :
This tutorial shows probability and counting and their important concepts. Tree diagrams and illustrations are used to show permutations in the tutorial examples. An example presents the Fundamental Counting Principle. Factorials and tree diagrams are use to show combinations in the tutorial examples.
The theorem of combination is presented in one of the examples to introduce the different probability distributions. The probability distributions are described in these examples. The graphical representation and tree diagrams are used to define probability and its use. The probability theorems are introduced with problems where they can be utilized. The probabilities of disjoint and overlapping events are presented in some of the tutorial examples.
Specific Tutorial Features:
• Several example problems with step by step illustrations of solutions.
• Tree diagrams are used to show counting and basic probabilities.
• Independent and dependent events are represented graphically.
• Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
• Definition slides introduce terms as they are needed.
• Visual representation of concepts
• Animated examples—worked out step by step
• A concise summary is given at the conclusion of the tutorial.
"Title" Topic List:
Counting Principle and Permutations Fundamental Counting Principle Permutations and their definitions Combinations Factorial Notation Combinations and their definitions Binomial Distributions and the Binomial Theorem The definition of probability and its use in applications problems Probability Theorems The probability of disjoint and overlapping Events Overlapping Events Theorem Disjoint Events Theorem Probability of Independent and Dependent Events