Sample Space (S) – set of all possible outcomes of a statistical experiment.

Example: When a coin is tossed.

Sample points – elements or members of the sample.

Statement or rule – description of sample space with infinite number of sample points or large number of sample points.

Example: S = {(x , y)/ x^{2} + y^{2} = 4}, S is a set of all points (x, y) such that these points are on the boundary or interior of circle whose radius is 4 and with center at the origin.

Event – subset of the sample space

The compliment of an event A with respect to S is the subset of all the elements of S that are not in A. The symbol is A’.

The intersection of 2 events A and B^{,}, denoted by the symbol A∩B, is the event containing all the elements that are common in A and B.

The union of 2 events A and B^{,}, denoted by the symbol A ᴜB, is the event containing all the elements that belong to A and B or both.

Two events A and B are mutually exclusive or disjoint if A ∩B = ɸ, that is if A and B have no elements in common.

Multiplication Rule – the fundamental principle of counting sample points

If an operation can be performed in n_{1 }ways, and if for each of the second operation can be performed in n_{2} ways, then the 2 operations can be performed together in n_{1}n_{2} ways.

Example: How many sample points are in the sample space when a pair of dice is thrown once?

n_{1 }= 6 ways

n_{2 }= 6 ways

n_{1} n_{2} = 36 possible ways

Generalized Multiplication Rule

If an operation can be performed in n_{1 }ways, and if for each of these a second can be performed in n_{2 }ways, and for each of the first two a third operation can be performed in n_{3 }ways, and so forth, then the sequence of k operations can be performed in n_{1 }n_{2 }. . . n_{k} ways.

Example: A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport and a patio or screened porch. How many different plans are available to this buyer?

n_{1 }= 4 ways

n_{2 }= 3 ways

n_{3 }= 2 ways

n_{4 }= 2 ways

n_{1.} n_{2 .} n_{3 . }n_{4 } = (4)(3)(2)(2) = 48 plans

Permutation – an arrangement of all or part of a set of objects

*The number of permutations of n objects is n!

*The number of permutations of n distinct objects taken r at a time is

Example: In how many ways can 6 people be lined up to get on a bus?

n = 6; n! = 6! = (6)(5)(4)(3)(2)(1) = 720 ways

Example: Two lottery tickets are drawn from 20 for first and second prizes. Find the number of sample points in the sample space S.

Circular Permutations – permutation that occur by arranging the objects in a circle. The number of permutations of n distinct objects arranged in a circle is (n – 1)!

Example: How many ways can 5 different trees can be planted in a circle?

n = 5; (n – 1)! = (5-1)! = 4! = (4)(3)(2)(1) = 24 ways

The number of distinct permutation of n things of which n_{1} are of one kind, n_{2} of a second kind, . . . n_{k} of nth kind is

Example: How many different ways can 3 red, 4 yellow, and 2 blue bulbs be arranged in a string of Christmas tree lights with 9 sockets.

n_{1} = 3 red bulbs

n_{2} = 4 yellow bulbs

n_{3} = 2 blue bulbs

n = 9 sockets

The number of ways of partitioning a set of n objects into 1 cell, with n_{1}, elements in first cell, n_{2} elements in the second, and so forth is

where

n_{1}+ n_{2}+ . . . + n_{r} = n, and r = number of cells

Example: in how many ways can seven scientists be assigned to one triple and two double hotel rooms?

The number of combinations of n objects taken r at a time is

Example: A printed circuit board may be purchased from five suppliers. In how many ways can three suppliers be chosen from the five?

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