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 A and B 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 B has occurs, then A occurs.
Theorem of Independent Events:
If A and B are independent events then and .
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"Title" Tutorial Summary :
This tutorial shows the principal concepts of probability and counting. Permutations are shown in the examples with the use of tree diagrams and illustrations. The Fundamental Counting Principle is described with the use of an example. Combinations are shown with the use of factorials and tree diagrams.
The theorem of combination is shown with the use of examples. The different types of probability distributions are described in this tutorial. The definition and use of probability in application problems are presented in this tutorial with graphical representation and tree diagrams. The probability theorems are introduced with scenarios where they can be utilized. Disjoint and overlapping events are presented along with their probabilities.
Specific Tutorial Features:
• Several example problems with step by step illustrations of solutions.
• Counting and Basic Probabilities are shown in the example with tree diagrams
• Independent and dependent events are given with graphical representations.
• 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 Definition of Permutations Combinations Factorial Notation Definition and formula of combinations Binomial Distributions and the Binomial Theorem The Definition and the use of probability Probability Theorems Probability of Disjoint and Overlapping Events Theorem I of Overlapping Events Theorem II of Disjoint Events Probability of Independent and Dependent Events