Textbook Notes (363,212)
MATH 203 (6)
Chapter 3

# Chapter 3 notes

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School
McGill University
Department
Mathematics & Statistics (Sci)
Course
MATH 203
Professor
Patrick Reynolds
Semester
Fall

Description
Math 203: Chapter 3 Sample Points Experiment – the process of observation that leads to a single outcome that cannot be predicted with certainty Sample point – basic outcome of an experiment Sample space – collection of all sample points of an experiment The probability of a sample point is a # between 0 and 1 which measures the likelihood that the outcome will occur. Example: Consider the experiment of tossing 2 coins. One sample point, for example, is (“Observe a head” and “Observe a tail”) Tree diagram: Assigning Probabilities The probability of a sample point could be taken to be the relative frequency of the occurrence of the sample point in many repetitions of the experiment. (theoretical) Of course, this may not be possible in progress. Subjective probabilities might be assigned based on general information of expert analysis of the experiment (e.g. weather). Events Let pirepresent the probability of sample point i. All sample point probabilities must lie between 0 and 1, 0 ≤ p i 1. The sum of probabilities of all sample points in a sample space must be 1: ∑ An event is a collection of sample points. To calculate the probability of an event, sum the probabilities of the sample points in the event. (back to example) Let’s suppose the coins are not balanced. The following probabilities are assigned to each of the sample points: HH | HT | TH | TT | Consider the event: “Observe exactly one head” (denoted by A) “A” consists of the sample points: HT and TH. The probability that A occurs is: P(HT)+ P(TH) = Consider the event: “Observe at least one head” (denoted by B) The event B consists of the sample points: B = {HH, HT, TH}. The probability B occurs is: P(HH) + P(HT) + P(TH) = Or ( ) Combinations Rule Suppose a set of n elements is to be drawn without replacement from a set of N elements. The number of different possible samples is denote( ) and is equal to ( ) ( ) Where: ( )( ) ( )( )( ) **Note: 0! Is defined to be 1 Ex: supposed you want to select 2 people from a group of 4. There are 6 different ways to do so. ( ) – pronounced “N choose k” ( ) ( ) ( ) Ex: lottery – pick 6 numbers without replacement from 53. You can do this( ) ways R: > choose(53,6) [1] 22957480 Unions, Intersections, Complements  Consider event A and event B Union: the event that occurs if either A or B (or both) occurs. Denoted by (AUB). - Venn diagram ** event – a collection of sample points - Union is all sample points in A, B, or both Intersection: the event that occurs if both A and B occur. Denoted by (A∩B) Example: Consider the experiment of rolling a fair die. Let A be the event “Roll and even number”. Let B be the event “Roll a number ≤3”.  A={2, 4, 6}  B = {1, 2, 3} C  A C{1, 3, 5}  B ={4, 5, 6}  AUB= {1, 2, 3, 4, 5, 6}  (A∩B)C= {2}  AUB ={5}  (A∩B) ={1, 3, 4, 5, 6} So each sample point in the sample space, S={1, 2, 3, 4, 5, 6} is equally likely, with probability  P(AUB)=  P(A∩B)= **Complement of A: the event that A does not occur. Denoted by A C Rule of CoCplements: P(A) + P(A ) = 1 (Equivalently) P(A) = 1- P(A ) Additive Rule: ( ) ( ) ( ) ( ) Mutually Exclusive Events: P( ) (equivalently) ( ) ( ) ( ) - Mutually exclusive events are 2 events that cannot happen at the same time (i.e. flipping a coin – the outcome can either be heads OR tails, NOT both) Example: Consider tossing 10 balanced coins. Let A be the event: A: “Observe at least 1 head” First, understand the sample space Basic sample point (example: HTHHTTHHHT, TTTTTTHHTH Here’s how we might count all the sample points: 2 2 2 2 2 2 2 2 2 2 = 2 = 1024 10 So thereCare 2 = 1024 sample points all equally likely. Note, A ={T T T T T T T T T T} C And P(A )= P(A) = 1-P(A ) = = Additive Rule (ex:) Reconsider the die experiment:  A={2, 4, 6}  B = {1, 2, 3}  P(A)=  P(B) =  A B={1, 2, 3, 4, 6}  A⋂B={2}  P(A B) = P(A) +P(B) – P(A⋂B) = Mutually Exclusive Events - No intersection - P(A⋂B)=0 Conditional Probability (3.5 textbook) Reconsider rolling a fair die  A = {roll even number}  B = {roll a number ≤3}  P(A)= But suppose we know that event B occurred. What’s the probability of A, given that B
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