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# 471 Review Sheet for MATH M0050 at Purdue

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Date Created: 02/06/15

Stat 225 Topic 1 Probability and Sets Introduction to Probability Think Physics not Math A common mistake made by many students who haven7t studied probability or statistics before is to think of it as a math course Probability is an application of math like physics not a particular type of it We will be using math in this course but none of it should be new The hard part of probability is learning when to apply the tools you will learn Of course if you do struggle with the math please come to of ce hours and get help The Purpose of Probability Let7s continue the comparison between probability and physics for another minute Physics uses math to model certain types of behavior in the real world For example if you were to drop a coin from a speci ed height you could calculate the rate at which it dropped using the principlestools you learn in physics ln physics the end result is always the same provided the initial conditions are the same same height same vacuum etc Probability allows us to study situations where the end result varies even when the initial conditions are the same or at least close enough to the same that we cannot distinguish one from another Even if we dont know exactly what the outcome will be probability allows us to make predictions on what the outcome is likely to be For example o If we ip a fair coin 20 times is it likely that it comes up a heads every time o If two parents have a daughter what should they expect the adult height of their child to be ls the answer likely to change if they have a son What if the sex of the child is unknown if for example the baby hasnt been born yet o If the stock market has been increasing every day for many days does that mean that it is likely to increase or decrease tomorrow o If children are picked at random for a part in the school play how likely is it that two best friends are both picked for major roles By the end of this course you should know how probability can be used to nd answers to all of these questions and many others Course Notation Notation is often one of the trickiest but most important parts of a learning a new subject lmagine trying to make sense of the formula A 7T gtlt r2 the formula for the area of a circle without knowing that 7139 is used to represent irrational number 3141592654 If you do not recognize a certain notation make sure to ask its easy to forget new notation a day or two after its been introduced and it can be hard to understand the topic of the current day when you7re trying to remember what the notation means Review of Sets and Set Notation Set De nition Notation 0 Sets are generally labeled using letters such as 0 An element of a set is represented by letters such as 0 To say that a is an element of the set A we write 0 To de ne a set we need to describe exactly which elements belong to it There are several ways to do this depending on what kinds of elements belong to the set Nonnumeric Sets Non numeric sets list each distinct element within Objects with multiple copies are only listed once For example what is the set of coins made by the US government for circulation When there are a large number of distinct elements it is sometimes possible to use shortcuts to avoid listing every element For example think of a short cut to describe the cards in a standard deck of 52 playing cards Countable Numeric Sets The principles from non numeric sets can always be used on countable numeric sets However countable numeric sets often contain pat terns that can be used to avoid listing every element separately 1 When there are a small xed number of elements in the set we list all ele ments in the set between as in the non numeric case Consequently set of possible results when a 6 sided die is rolled is 2 When there is a countably in nite number of numeric elements displaying a clear pattern we list enough elements at the beginning to establish the pattern and replace the remaining elements with dots The natural counting numbers can be written as Remember that a countably in nite set has in nitely many elements but in theory all of them can be listed 3 When there is a nite number of numeric elements displaying a clear pattern in addition to establishing the pattern it is necessary to show where to stop Thus the last element is listed after the dots Even numbers from 2 to 100 can therefore be written as Uncountable Numeric Sets Remember that uncountable sets are impossible to list For example try to list the numbers from 0 to 1 After 0 no matter how small of a number you choose to come next you can always nd one smaller Since listing the elements of these sets are impossible the idea is to instead focus on where the maximum and minimum for a continuous range are well talk about how to work with multiple continuous ranges momentarily A is used to denote a boundary that includes the boundary point A is used to denote a boundary that does not include the boundary point or goes to in nity or negative in nity Examples 1 The range of numbers from 0 to 1 including 1 but not 0 2 All non negative numbers Special Sets There are two special kinds of sets that we need to make note of The empty set Subset Working With Sets Consider two sets A B There are 3 basic operations we can perform on these sets to create new sets 0 To create a new set that consists of all elements in either set we take the 0 To create a new set that consists of the elements in both sets we take the 0 To create a new set that consists of all elements not in set A or B we take the Note Using the compliment requires some knowledge of what the elements not in A are This should be clear from the contest of the problem Laws about Sets Let AB and C be sets Then the following laws are always true Commutative Law Associative Law A BB A A B OA B O AUBBUA AUBUOAUBUO Distributive Law De Morgan s Law A BUOA BUA O AUBCAC BC AUB OAUB AUO A BCACUBC Visualizing Sets Venn Diagrams provide a simple method for Visualizing relationships between sets Union A U B Intersection A B Complement Ac Examples LetA 110B 24207 andO 14710713716719 LetU 120 be the entire set 7 ie Ac U Ac 117 7 20 Determine the elements of the following sets 1 Ac 2 AUB 3 A O 4 O AUBC 53mmm0 aimummm 1A BVUB Draw a Venn Diagram to represent each of the following situations 1 A is a subset of B7 and A is a subset of C 2 AisasubsetofB B Cy Q A C 3 A and B are subsets of Cc Sets and Probability Random Experiment De nition Examples Sets in Probability Sets are used in probability to label groups of possible results of a random experiment Some important sets used in probability include Sample Space Event Outcome Examples Identify the sample space7 the desired event7 and the outcome 1 I ip a coin three times looking for at least 2 heads I actually get a head7 then a tail7 then a second tail 2 I roll a red and a white dice7 looking for the sum to be 10 I actually roll a red six7 and a white four 3 I draw a card from a 52 card deck7 looking for a face card I actually get the ace of diamonds 4 I draw a 5 card poker hand from a 52 card deck7 and l7m looking for a ush all 5 cards are from the same suit Mutually Exclusive Events De nition Note Always keep an eye out for mutually exclusive events they can make many problems much easier Example Are the following events mutually exclusive 1 Getting an even number and getting a number lower than 4 when I roll a 6 sided die 2 If I roll a die two times7 getting an even number on the rst roll7 and the sum of the two dice being 2 3 If I roll a die two times7 getting an even number on the rst roll and getting an odd number on the second roll Frequentist De nition of Probability De ne PE to be the probability that an event E occurs when a given random experiment is performed Suppose that this random experiment is repeated 71 times7 where n is a large number Let be the number of times that the event E occurs Then the frequentist interpretation of probability states that Note PE is a function which takes a set E and translates into a number Axioms of Probability There are three basic assumptions about the function PE needed for probability to make sense These are known as the Axioms of Probability7 or the Nonnegat ivity Certainty Additivity Calculating Probabilities When each element in a sample space is then we can calculate the probability of an event by the probabilities of each element in the event In other words7 let NE be the number of elements in the set E Then Example Suppose a random experiment consists of rolling a blue and a red dlie7 and recording the sum of both dice 1 How many different numbers can the two dice sum to 2 How many equally likely outcomes are there 3 What is the probability that the sum of the dice is 5 4 What is the probability that the sum of the dice is greater than or equal to 9 5 What is the probability that the sum of the dice is even 6 What is the probability that the sum of the dice is not 67 77 or 8 10 Properties of Probabilities Probability of the Empty Set Domination Principle Complementation Rule Note The complementatz39on rule is one of the most useful properties in probability Law of Partitions A partition is a group of events7 A1 An in the same sample space where the of all the events is equal to the sample space The Law of Partitions states that the probability of an event B can be found by General Addition Rule When A7 B7 and C are not mutually exclusive7 the probability of their union is Pmum PAUBUO This can be generalized to an arbitrary number of sets using the same principle 11 Examples 1 A survey is sent to a random sample of 50 undergraduates at Purdue with 3 yesno questions Four people responded that they exercise once a week7 are from lndiana7 and have a family member who has also attended Purdue Six people exercise once a week and have a family member at Purdue Seven people are from Indiana and have a family member who has attended Purdue Seventeen people have a family member who has attended Purdue Eight people from Indiana responded that they dont exercise7 and 8 people who exercise are not from Indiana Sixteen people responded no to all three questions a Complete a Venn diagram displaying the number of individuals in each category b If we were to put all the responses into a hat7 what is the probability that i We draw out a response that has a yes for all questions ii Draw out a response with exactly two yeses iii Draw out a response with less than two yeses iv What is the probability we get a response of someone who is from lndiana7 or has a family member who has attended Purdue v What is the probability that the respondent answered yes for at least one ques tion 12 2 Shade the regions of a 3 way Venn Diagram corresponding to the following regions a AUB b AUB O C A BV C d Cu B m AU 00y 13 3 If F is the event that a randomly chosen person is female7 and S is the event that the person is single7 then how would you describe the events that the randomly chosen person is a Married b A married female c A single male 4 In a class with 100 students7 50 like math and 60 like statistics 15 like both math and statistics Draw a Venn diagram and answer the following questions a How many students like only math b How many like neither math nor statistics 14 5 Suppose you roll two fair six sided dlice7 one red and one blue a What is the probability that the two dice will show the same face b What is the probability that the two dice will show different faces C What is the probability that the blue dice shows a 37 and the red die a 4 d What is the probability that one of the dice shows a 37 and the other shows a 4 15

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