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# Math 170 Exam 1 study guide MATH 170 001

USC

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This 4 page Study Guide was uploaded by Savannah Notetaker on Sunday September 25, 2016. The Study Guide belongs to MATH 170 001 at University of South Carolina taught by Gregory Clark in Fall 2016. Since its upload, it has received 26 views. For similar materials see Finite Mathematics in Math at University of South Carolina.

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Date Created: 09/25/16

Math 170 Exam 1 Chapter 6 Definitions : Set- a set is a collection of items, which we can call elements. Notation- a set is typically denoted by a capital letter, it’s elements are separated by commas, and the elements begin and end with brackets o Ex: A= {1,2,3} Famous set: o Natural numbers o The integers o The empty set Union- the union of 2 sets A and B is denoted A U B o Ex: A= {1,2} B= {3,4} …. A U B = {1,2,3} Intersection- the intersection of 2 sets A ∩ B is A ∩ B = {X: x E A and x E B} o Ex: R= {a,b,1,2} S= {a,2,3} ….. R ∩ S= {a,2} The Cartesian product- we call R x S the cartesian product of R and S o Ex: A={1,2} B= {x,y,z} ……. A x B= { (1,x) (1,y) (1,2) (2,x) (2,y) (2,2) o Ordered pairs Addition principle- when choosing between r disjoint alternative, suppose alternative one has n, possible outcomes, alternative 2 has n2 … etc o With no two of these outcomes the same… then the total number of possible outcomes is n1 + n2 +… + nr Multiplication principle- when making a sequence of choices with r steps suppose that step one has n1 choice, step 2 has n2 , step r has nr. And that each sequence of choice results in a different outcome, then the total number of outcomes is n1 x n2 x n3 x …nr Decision algorithm- a story which counts the possible outcomes of a scenario. o Ex: how many diff. tacos are possible given we choose one ingredient from all 3 categories? o – use mult. Principle o 5 x 5 x 8= 200 Permutation- a permutation of n distinct elements is an ordering of the elements o In general the number of permutation n distinct elements is n! o 0! = 1 o Order matters Combinations- the number of ways to form a K element set from an n element set is : n choose K o n!/ k! (n-k)! o Order doesn’t matter- unordered Chapter 7: Experiment- an occurrence with a result Outcome- a result of an experiment Sample space- is the set of all outcomes o Ex: flip a fair coin S={H,T} Event- an event is a subset of the sample space. Elements of the event are called favorable o Ex: experiment rolling a 6-sided die o S={1,2,3,4,5,6} o E= {2,4,6} o F={1,2,3} o E U F= {2,4,6,1,3} o Two events are mutually exclusive if they have empty intersection Experimental probability- “real life probability”, you have an experiment, keep track of results in a sample space. Want to find the experimental probability of an event. Frequency- the frequency of an event E ≤ S is the number of times E occurs during the experiment. Denoted fr(E) The experimental probability of an event E is PE(E)=fr(E)/N (total # of trials) o Ex: flip H/T coin with results: outcomes: H T # of times: 7 3 Q: what is N? N=fr(H) + fr(T) = 7 + 3 =10 o PE({H})= fr({H})/N = 7/10 PE({T})= 3/10 PE({H,T})= fr({H,T})/N = 10/10 Experimental probability distribution- is a table of the events of an experiment and the experimental probability of each. o Properties of an E.P.D.: o PE(E) ≥ 0 o PE(E) ≤ 1 o PE(S1) + PE(S2) + PE(S3) + … PE(Sn) = PE({S1,S2,S3,…Sn}) Theoretical probability- o Formula: P(E) = │E│/│S│ o When all outcomes are equally likely use theoretical formula o Ex: Q: if I flip 2 distinguishable fair coins, what’s the probability I flip 2 heads? S= {HH,HT,TH,TT} P({HH})= │{HH}│/ │S│= ¼ Probability distribution- use the chart Conditional probability- “given” the conditional prob. of “A given B” is the probability that A occurs given B has occurred. o P(A l B)= P( A ∩ B)/ P(B) o P(B) cannot = 0 o Ex: roll a fair 6-sided die A= rolling a 4 B= rolling an even P(A l B)= 1/6/1/2= 1/3 Baye’s Theorem- o P(A l T)= P(T l A) P(A)/ P(T l A) P(A) + P(T l A’) P(A’) o Very long questions o Will always present the info. “A given B”

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