elemprob-fall2010-page1

elemprob-fall2010-page1 - A is 1 / 36 times the number of...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 3160, Lecture 3 1 Framework We start with a sample space Ω, a class of events, and a probability. The sample space can be any set whatsoever. An event is a subset of the sample space. Occasionally we restrict the collection of subsets to what is called a σ -field, but most often we allow any subset of Ω to be an event. To each event A we assign a number P ( A ), the probability of A , which is a number between 0 and 1. As an example, suppose we roll a die. (“Die” is the singular form of the plural “dice.”) We set Ω = { 1 , 2 , 3 , 4 , 5 , 6 } and the set of events is the collection of all subsets of Ω. We assume the die is fair, and we set P ( A ) to be equal to the number of elements in A divided by 6. Thus P ( { 1 , 3 , 5 } ) = 3 6 = 1 2 . For another example, suppose we roll two dice, one green and one red. Then we take Ω to be all pairs (1 , 1) , (1 , 2) ,..., (6 , 6) and the probability of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: A is 1 / 36 times the number of elements in A . In both these cases, we assume that each number on each die is equally likely. We might have a loaded die, where P ( { 1 } ) = . 10, P ( { 2 } ) = . 15, etc. We do want something to happen, so P () = 1 and P ( ) = 0. The collection of events have to be what is called a -eld. A collection F subsets of is a -eld if , F , A c F whenever A F , and i =1 A i and i =1 A i are in F whenever all the A i are in F . A probability is a function on the events such that (1) 0 P ( A ) 1 for each event A . (2) P () = 1 and P ( ) = 0. (3) If A and B are disjoint, which means A B = , then P ( A B ) = P ( A ) + P ( B ) . 1...
View Full Document

This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.

Ask a homework question - tutors are online