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PRED 35 4 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

PRED 35 4 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH. Lesson 6 Conditional Probability. Question.

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PRED 35 4 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

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  1. PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH Lesson 6 Conditional Probability

  2. Question Suppose that a certain precinct contains 350 voters, of which 250 voters are Democrats and 100 are Republicans. If 30 voters are chosen at random from the precinct, what is the probability that exactly 18 Democrats will be selected?

  3. Question If A, B, and D are three events such that what is the value of

  4. Question Iftheletterss, s, s, t, t, t, i, i, a, c,arearranged in a randomorder, what is theprobabilitythattheywillspelltheword “statistics”?

  5. Conditional Probability The updated probability of event A after we learn that event B has occurred is the conditional probability of A given B. Conditional probability of A given that B has occurred.

  6. Example Findtheprobability of a 1, giventheoccurrence of an oddnumber, in thetoss of a singledie.

  7. Independence Two events, A and B, are said to be independent , if Otherwise, the events are said to be dependent.

  8. Independence - Disjoint events Two events, A and B, are said to be independent , if Two events are disjoint if

  9. Example Considerthefollowingtwoevents in thetoss of singledie. A: observe an oddnumber B:observe an evennumber C:observe a 1 or 2. • Are A and B independentevents? • Are A and C independentevents?

  10. Example Threebrands of coffee, X, Y and Z, areto be rankedaccordingtotasteby a judge. Define thefollowingevents: A: brand X is preferredto Y, B: brand X is rankedbest, C: brand X is rankedsecondbest D: brand X is rankedthirdbest. Ifthejudgeactually has no tastepreferenceandthusrandomlyassignsrankstothebrands, is event A independent of events B, C, and D?

  11. Two laws of Probability Additive law: The probability of union of two events A and B is If A and B are mutually exclusive events,

  12. Two laws of Probability Multiplicative law: The probability of the intersection of two events A and B is If A and B are independent,

  13. Two laws of Probability Multiplicative law: Prove the probability of the intersection of any number of events

  14. Bayes’ Rule Suppose that the events form a partition of the space and for Then, for every event A in S,

  15. Example Supposethat a personplays a game in which his scoremust be one of the 50 numbers 1, 2, …., 50 andeach of these 50 numbers is equallylikelyto be his score. Thefirst time he playsthegame, his score is X. He thencontinuestoplaythegameuntil he obtainsanotherscore Y suchthat Y≥X. Assumethatallplays of thegameareindependent. Determinetheprobability of theevent A that Y=50.

  16. Example An electronicfuse is producedbyfiveproductionlines in a manufacturingoperation. Thefusearecostly, arequitereliable, andareshippedtosuppliers in 100-unitlots. Becausetesting is destructive, mostbuyers of thefuses test only a smallnumber of fusesbeforedecidingtoacceptorrejectlots of incomingfuses. Allfiveproductionlinesnormallyproduceonly 2% defectivefuses, whicharerandomlydispersed in theoutput. Unfortunately, productionline 1 sufferedmechanicaldifficulty 5 % defectivesduringthemonth of March. Thissituationbecameknowntothemanufacturerafterthefuses had beenshipped. A customerreceived a lot produced in Marchandtestedthreefuses. Onefailed. What is theprobabilitythatthe lot wasproduced on line 1? What is theprobabilitythatthe lot camefromone of thefourotherlines?

  17. RANDOM VARIABLE

  18. Random Variable A real-valued function that is defined on the space S is called a random variable. The probability that X takes on the value of x, Pr(X=x), is defined to be the sum of the probabilities of all sample points in S which are assigned the value x by the function X. EX: Tossing a coin: A coin is tossed 10 times. Let X be the number of heads that are obtained.

  19. Probability distribution The set of all pairs for which is called the probability distribution for X. EX: A foreman in manufacturing plant has three men and three women working for him. He wants to choose two workers for a special job. Not wishing to show any biases in his selection, he decides to select the two workers at random. Let X denote the number of women in his selection and find the probability distribution for X (as histogram)

  20. Binomial distribution A binomial experiment is one that possesses the following properties: 1. The experiement consists of n identical trials. 2. Each trial results in one of two outcomes. For lack of a better nomenclature, we call one outcome a success, S, and the other a failure, F. 3. The probability of success on a single trial is equal to p and remains the same from trial to trial. The probability of failure is equal to 1-p=q. 4. The trials are independent. 5. The random variable of interest is X, the number of successes observed during the n trials.

  21. Binomial distribution

  22. Binomial distribution Ex1: A coin is tossed ten times. Probability of observing seven times 1 or 2 Ex2: Consider a family having six children. Probability of observing two boys

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