ࡱ> VXU r:bjbjcTcT 4D>>Z::}}$\-|]^   $-&-&-&-&-&-&-E02&-] ^   &-}}-6"6"6" 8}$-6" $-6"6"r\,Tk^,B!,--0-,236"3,3,  6"     &-&-6"   -    3         : C:  Exemplar 1 Graham and Samantha play a game of cards. They each have a set of cards numbered 1 to 10. They call themselves player A and player B. One game consists of each player selecting a card at random from their set. They play twenty games, recording the results. Player A wins if both cards are the same. Player B wins if the total of the two cards is 12. Graham and Samantha must decide how many points each player should be awarded when they win a game to make the game fair. Using a table or a tree diagram to systematically show all possible outcomes they are able to calculate the probability of a win for each player and work out a marking system. Could try other variations, for example player B wins if the cards are different. Exemplar 2 Graham and Samantha begin to extend their understanding of probability by considering the probability of combined events. Members of the class write their name on the correct region of a Venn Diagram, showing which of the option subjects Bio or History they take. Teacher then reduces the diagram to the number of students in each region.  Rhea and Sarah discuss the probability of a student taking both subjects, or bio or history. They become aware of the ambiguity of this question, and work out the probability of each of the two meanings, inclusive of both subjects or not. Teacher leads them to use set notation to describe their findings using set notation, and for the inclusive or P(B U H) = P(B) + P(H)  P(B)"H) For some students it is better to continue to use probability language such as AND and OR rather than set notation. Exemplar 3 Graham and Samantha take part in a class discussion about birthdays. The teacher asks the students to consider this: If you ask for the birthday of someone in the classroom how likely is it that it will be in January? The teacher leads them to the idea that they can get a first cut at an answer by making some assumptions (eg all days of the year have equal number of births in New Zealand). There is data on births which would give a better estimate. If you know that pizza is the favourite food of the person you are asking, will this affect how likely it is that this students birthday is in January? Why or why not? If you know the country of birth will that affect it? From the discussion Graham and Samantha develop an informal idea of independence: that the likelihood of a particular birth month is probably not related to the favourite food of the person. The teacher then leads them to the idea of formal independence of events by allowing them to calculate probabilities from a two-way table of their option subject data. They notice that the probability of a student taking Bio depends on whether the student takes history or not. (link to Exemplar 2) Takes bioDoes not take bioTakes history51217Does not take history8715131932 Conditional probability The probability that a randomly chosen student who takes history also takes bio is 5/17 = 0.294 The cell values used in this calculation are Number of students who take bio and history Number of students who take history This can be written as Number of students who take bio and history | total number of students Number of students who take history | total number of students Written in terms of probability this is P( takes Bio and History) P(takes History) Using set notation this is P(B)"H) P(H) Graham and Samantha continue to evaluate other conditional probabilities from the table. ****Fred thinks that P(H | B) and P(B | H) are obviously the same. Investigate. Exemplar 4 By attempting to draw a probability tree diagram of the equivalent to the two way table to calculate the probabilities of taking bio and history options, Graham and Samantha notice that multiplying along the branches does not produce the correct probabilities. They realise that this occurs because the events B and H are not independent. They turn their attention to tossing two dice, one red, one blue, considering the two events Event A: a five turns up on the blue die Event B: a three turns up on the red die. They consider whether these two events are independent, realising that we would not expect the outcome of tossing the red die to affect the outcome on the blue die. They calculate the probability of event A occurring, event B occurring and both events occurring simultaneously P(A INCLUDEPICTURE "http://people.hofstra.edu/stefan_waner/realworld/tutorialsf3/gf/intersect.gif" \* MERGEFORMATINET B), and notice that P(A). P(B) = P(A INCLUDEPICTURE "http://people.hofstra.edu/stefan_waner/realworld/tutorialsf3/gf/intersect.gif" \* MERGEFORMATINET B). They then consider Event T: the total on the two dice is eight, and whether events A and T are likely to be independent, that is, is the blue die is likely to affect the probability of the total score. They calculate P(T) = 5/36, and P(A INCLUDEPICTURE "http://people.hofstra.edu/stefan_waner/realworld/tutorialsf3/gf/intersect.gif" \* MERGEFORMATINET T) = 1/36 and note that P(A). P(T) `" P(A INCLUDEPICTURE "http://people.hofstra.edu/stefan_waner/realworld/tutorialsf3/gf/intersect.gif" \* MERGEFORMATINET T). 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HYPERLINK "http://www.stats.govt.nz/" http://www.stats.govt.nz/ For example: employment data Look at ethnicity and whether they are employed or not. Find P(Employed/Ethnicity=X). Do employed and ethnicity seem to be independent. Could also look at health data, type of diet. 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[ @Verdana;(SimSun[SOC+H Lucida Grande?= *Cx Courier New;WingdingsA BCambria Math"h&&fV (V (!n724dOOr2qHX)?2! xx8Mathematics and Statistics in the New Zealand Curriculumstaffpipa    Oh+'0x  ( 4 @ LX`hp<Mathematics and Statistics in the New Zealand CurriculumstaffNormalpipa2Microsoft Office Word@F#@KO@@V՜.+,D՜.+,x4 hp  Ministry of Education( O 9Mathematics and Statistics in the New Zealand Curriculum Title 8@ _PID_HLINKSAl  http://www.stats.govt.nz/=  !"$%&'()*,-./0123456789:;<=>?@ABCDFGHIJKLNOPQRSTWRoot Entry F0BYData #1Table+3WordDocument4DSummaryInformation(EDocumentSummaryInformation8MCompObjy  F'Microsoft Office Word 97-2003 Document MSWordDocWord.Document.89q