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Chapter 13 . . event spacesample is called the called is is a situation in which we know what outcomes could could what outcomes which we know in a situation is all possible outcomes all possible . . trial called a called happen, but we don’t know which particular outcome did or will happen. will or did know which outcome we particular happen, don’t but random phenomenon random Dealing with Random Phenomena Random with Dealing The collection of of The collection At each trial, we note note the the random phenomenon,value eachweAt trial, an of it call and an is combination the resulting When combine outcomes, we A a randomobserve phenomenonIn general, which we upon eachis occasion

ON TARGET It can be written as a fraction, decimal, a fraction, as be written It can ratio. or percent, The greater the probability the more more the the probability greater The will occur. the event likely It tells us the relative with which we can we which with frequency us It tells the relative to occur event an expect Probability is the of chance. is the mathematicsProbability

ON TARGET 50/50 Certain Impossible 0 1 .5 Definitions Sum of Sum of the of all events is 1. Value is between 0 and 1. 0 and is between Value of of the likelihood measure will occur. event that the Probability is the numerical Probability

ON TARGET , denoted as S. as S. , denoted in a probability probability in a trial Definitions . The of all The set possible outcomes of a probability is the experiment All 6 faces{ 1 , of a die: S = 2 , 3 , 4 , 5 , 6 } e.g. outcome is an The result single of a The result A probability experiment is an action through which through is action an probability experiment A responses) results or (counts, measurements, specific are obtained.

ON TARGET these. Definitions space. May use a combination of a combination use May Diagrams Where appropriate always display your your displayalways Where appropriate sample Tree Tree Lists Diagrams Lattice Diagrams Venn Other Examples of Sample Spaces may include: may SampleExamplesOther Spaces of

ON TARGET Definitions consists of one or more outcomes and is is a and outcomes one or more of consists Notation: The probability that event E will event probability that The Notation: is read P(E) is and written occur E.” “the of event probability event letters, such as A, B, or C. letters, such as Events are often represented by uppercase represented are often Events subset of of the samplesubset space. An

ON TARGET to occur. to occur. (There are 2 waysto (There are 2 and the 6 one get 4) other 1 18  equally likelyequally 2 36 and of the Outcomes in the Sample Space in the Sampleof the Outcomes Definitions random Each are Number of of Event Outcomes Number Total Number of Possible in S Outcomes of Possible Number Total e.g. e.g. P( ) = The Probability The Event, of E: an P(E) P(E) = • Consider a pair of a pair Consider •

ON TARGET  probability formula probability formula Number of Event Outcomes of Event Number Definitions Total Number of Possible in S Outcomes of Possible Number Total Ultimate P(E) = The in a sample space is equally likely to occur. likely is a sampleequally in space 1. Theoretical Probability Theoretical 1. outcome each when probability is used Theoretical There are three types of probability of types three are There

ON TARGET results, results, or possible . Probability This kind of probability This kind probability of is referred to as theoretical probability Probabilities determined using mathematical Probabilities computations based on outcomes. Theoretical Theoretical

ON TARGET Number of Occurrences Number Event Total Number of Observations Number Total Probability (or (or Probability Probability) Empirical Definitions P(E) P(E) = E is the relative frequency of event E of event frequency E is the relative The experimental probability of an probability event The experimental 2. 2. Experimental There are three types of probability of types three are There obtained from from probability . obtained Experimental probability is based upon observations observations probability is based upon Experimental

ON TARGET experimental experimental . Also known as Relative Frequency. Frequency. Also known as Relative probability. probability kind This of is referred as to probability experimentation experimentation observation, and results, recording and then using these results to predict expected Probabilities determined from repeated Probabilities Experimental Probability Experimental

ON TARGET Definitions Probability probability is a probability is a Therefore, there is no formula to calculate it. to calculate is no formula there Therefore, Usually found by consulting an expert. expert. an by consulting found Usually Personal Personal resulting from intuition, educated guesses, guesses, and educated from intuition, resulting estimates. 3. 3. Personal There are three types of probability of types three are There

ON TARGET run - States States that the long Roughly speaking, this that the that means this speaking, Roughly - The Law of Large . of Large Law The Independent Independent the changeor influence doesn’t trial one of outcome another. of outcome For example, coin flips are independent. are flips coin example, For increases. relative relative frequency (experimental probability) of repeated independent events gets closer and closer to the theoretical probability as the number of trials Related by Related Numbers: of Large The Law Theoretical vs. Experimental Experimental vs. Theoretical Probability

ON TARGET Definitions Numbers The greater the number of the number trials the more The greater probability of an the experimental likely its theoretical equal will event probability. Law of Large Large of Law the theoretical probability of the event. of probability the event. the theoretical As an experiment is repeated over and over, the over, and over is repeated experiment As an approaches probability event of an experimental

ON TARGET Example: Theoretical vs. Experimental Experimental vs. Theoretical Example: Probability In the long long run, In the if many you flip a times, coin heads will occur of the time. about ½

ON TARGET occur occur actually should What – What – occurred or happened occurred or (relative frequency). or happen. or Experimental Probability Theoretical Probability Theoretical vs. Experimental Experimental vs. Theoretical Probability

ON TARGET , long, long, in run behavior. - infinitely only only in long the run long long ( really “due” to occur) doesn’t exist at all. fact). of a (that an outcome Averages so called The Law of event random that hasn’t occurred in many trials is Relative frequencies even out Relative and this run long is The LLN saysThe about nothing short The Nonexistent of Law Nonexistent The

ON TARGET Make a picture. Make a picture. Make a picture. The First Three Rules of Working with with Working of Rules Three First The Probability data, the threebut rules don’t change. We are dealing with probabilities now, not not are dealing with probabilities now, We

ON TARGET Venn Diagrams Venn Diagrams Tree Lists Diagrams Lattice Number of of Event Outcomes Number Total Number of Possible Outcomes in S Outcomes of Possible Number Total Pictures: P(E) P(E) = space. space. all have Once possible been outcomes indentified, calculating probability the of an event is: All the pictures use we help us indentify sample the The First Three Rules of Working with with Working of Rules Three First The Probability

ON TARGET . ) ≤ 1 ) A ( P 0 ≤ 0 , A Probability (Laws of Probability) of (Laws Probability Two requirements requirements probability:a for Two For any event event any For A probability is a number between 0 and 1. and 0 between number is a probability A 1. Formal Formal

ON TARGET trial represents the set of all possible outcomes.) possibleall of set the represents be 1. S ( must Probability ) = 1 ) = S Probability Assignment Rule: Assignment Probability ( P The probability of the set of all possible outcomes of a outcomes possible all of set the of probability The 2. Formal Formal

ON TARGET is called ) C A A P( – ) = 1 ) A ( . in the event the in P C A not , denoted denoted , A of complement the the is 1 minus occurring event an of probability The occur: doesn’t it that probability The set of outcomes that are are that outcomes of set The Probability Complement Complement Rule: . . 3. Formal Formal

ON TARGET mutually mutually (or disjoint ). occur together) are called are together) occur exclusive Addition Addition Rule: Events that have no outcomes in common (and, thus, cannot thus, (and, commonin outcomes no have that Events 4. Formal Probability Formal

ON TARGET the or are disjoint. are B and A , the probability that one one that probability the , B and A provided that provided , ) B ( P ) + ) A ( P ) = Probability B other occurs is the sum of the probabilities of the two the of probabilities the of sum is the occurs other events.  A Addition Rule (cont.): Addition Rule ( For two disjoint events disjoint two For P 4. Formal Formal

ON TARGET are B and A , the probability that probability the , B and A provided that that provided , ) B ( occur is the product of the probabilities of probabilities the of product is the occur or change the outcome of another. of outcome the change or P the outcome of one trial doesn’t influence influence doesn’t trial one of outcome the . B  - ) A ( and P A ) = B both events. two the independent Probability  A Multiplication Multiplication Rule: ( Independent P For two independent events events independent two For 5. Formal Formal

ON TARGET are not disjoint, provided disjoint, not are B and A the two events have probabilities greater than zero: than greater probabilities have events two the Probability Multiplication Rule (cont.): Multiplication Rule Two independent events independent Two 5. Formal Formal

ON TARGET Independence Independence independence doesn’t make it make doesn’t independence assuming Rule (cont.): Rule , but but , about whether that assumption is reasonable is assumption that whether about Think Assumption true. Rule. Multiplication the using before Probability Multiplication Multiplication Always Many Statistics methods require an require methods Statistics Many 5. Formal Formal

ON TARGET ) B  A ( P ) Notation B ) B -   A ( A P ( P ). B  ) instead of ) instead B ) instead of ) instead A B ( P or and A A ( ( P P following: and In other situations, other situations, In see the you might Notation Notation alert: this text we the notation use In Formal Probability Probability Formal

ON TARGET of the of the complement combination. easier to work with the event we’re really interested in. probability, we’ll use in the rules probability, A good thing to remember is that can that it be to remember is thing good A In most situations most situations In want we where a to find Putting the Rules to Work to Rules the Putting

ON TARGET probabilities for all possible outcomes must total 1. total must outcomes all possible for probabilities Events must be disjoint to use the Addition Rule. Addition the use to disjoint be must Events To be a legitimate , the sum of the of the sum the distribution, probability a legitimate be To disjoint. Don’t add probabilities Don’t add probabilities of events they’re not if Beware of probabilities Beware probabilities of that 1. add don’t to up What Can Go Wrong? Go Can What

ON TARGET — . be independent can’t independent is one of the most common errors people people errors common most the of is one independent probabilities. with in dealing make The multiplication of probabilities of events that are not are that events of probabilities of multiplication The disjoint disjoint events not independent. Don’t confuse disjoint Don’t confuse disjoint and independent Don’t multiply probabilities multiply Don’t of events if they’re What Can Go Wrong? Wrong? Go Can What

ON TARGET - run relative relative run - Watch out for misinterpreting the LLN. the misinterpreting for out Watch run behavior. frequencies. Probability is based on long Probability Law The speaks of Large Numbers long only of What have we learned? we have What

ON TARGET Multiplication Rule for independent events independent for Rule Multiplication Probability Assignment Rule Assignment Probability Rule Complement events disjoint for Rule Addition probabilities of more complex events. We probabilities complex of more events. We have the: probabilities probabilities of outcomes find to There are some basic basic some are There for combining rules What have we learned? learned? we have What

ON TARGET 19 all, 19 – 13 all, 15 all, 15 13 - 360 - 341: #8 #8 341: – 15, pg. 342 pg.15, - 25 odd, 31 25 14, pg.338 14, – - Read Ch Read Ch 21 Assignment

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