Chapter 6 COUNTING AND PROBABILITY
6.1 Introduction
Random Process, Sample Space, Event And Probability
Random Process, when it takes place, one outcome from some set of outcomes is sure to occur, but it is impossible to predict with certainty which outcome that will be.
Sample Space, the set of outcomes from a random process.
Suppose that the deck of cards have been shuffled. Then considered the following problem.
Counting The Element of a List
6.2 Possibility Trees and the Multiplication Rule
The Multiplication Rule
The result of this example is that to count the number of tables(4 rows of 2input combination with one column of output)
When the Multiplication Rule is Difficult or Impossible to Apply
A Complex way to choose the officer.
Permutations
A permutation of a set of objects is an ordering of the objects in a row. For example ,the set of elements a, b, and c has six permutations.
abc acb cba bac bca cab
Permutations of Selected Elements
6.3 Counting Elements of Disjoint Sets:
The Difference Rule
The Inclusion/Exclusion Rule
6.4 Counting Subsets of a Set: Combinations
The phrase at least n means "n or more."
The phrase at most n means "n or fewer."
Some Advice about Counting
If you can imagine the elements to be counted as being obtained through a multistep process(in which each step is performed in a fixed number of ways regardless of how preceding steps were performed), then you can use the multiplication rule. The total number of elements will be the product of the number of ways to perform each step.
If, however, you can imagine the set of elements to be counted as being broken up into disjoint subsets, then you can use the addition rule. The total number of elements in the set will be the sum of the number of elements in each subset.
One of the most commmon mistakes is to count certain possibilities more than once.
6.5 r-Combinations with Repetition Allowed
Which Formula To Use
Four different ways of choosing k elements from n. the order in which the choices are made may or may not matter, and repetition may or may not be allowed.
Summarizes which formula to use in which situations:
6.6 The Algebra of Combinations
Pacal's Formula
6.7 The Binomial Theorem
6.8 Probability Axioms and Expected Value
Recall that the sample space is a set of all outcomes of a random process or experiment and that an event is a subset of a sample space.