A Lesson in Probability

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)/ x² + y² = 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 n₁ ways, and if for each of the second operation can be performed in n₂ ways, then the 2 operations can be performed together in n₁n₂ ways.

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

n₁ = 6 ways

n₂ = 6 ways

n₁ n₂ = 36 possible ways

Generalized Multiplication Rule

If an operation can be performed in n₁ ways, and if for each of these a second can be performed in n₂ ways, and for each of the first two a third operation can be performed in n₃ ways, and so forth, then the sequence of k operations can be performed in n₁ n₂ . . . 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?

n₁ = 4 ways

n₂ = 3 ways

n₃ = 2 ways

n₄ = 2 ways

n_1. n_{2 .} n_{3 .} n₄  = (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 n₁ are of one kind, n₂ 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.

n₁ = 3 red bulbs                                                                                                                                 

n₂ = 4 yellow bulbs

n₃ = 2 blue bulbs

n = 9 sockets

The number of ways of partitioning a set of n objects into 1 cell, with n₁, elements in first cell, n₂ elements in the second, and so forth is

  where

n₁+  n₂+ . . . + 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?