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The International School of Toulouse Department of Mathematics. 
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\par }{\b\i\f28\lang1033\langfe1033\langnp1033 Work done in this assignment will be assessed against the criteria A, B, C, D and F. You are therefore expected to use a graphic display calculator/computer for this assignment.
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\par }\pard \qc \li0\ri0\sa120\sl-320\slmult0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 {Alex has a coin with a '1' on one side and a '3' on the other. Beth's coin has a '2' on both sides. \line Each player begins with a starting value of  }
{\f1\fs20 1}{.\line They multiply the starting value by the number they throw on their respective coins.
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\par }\pard\plain \ql \li0\ri0\sa120\sl-320\slmult0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 \fs24\lang2057\langfe1033\cgrid\langnp2057\langfenp1033 {\b\ul\lang1033\langfe1033\langnp1033 Section One; The original game;
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After many games, do you expect that the outcome will be: \line          (a) Alex gets a higher score than Beth?\line          (b) Beth gets a higher score than Alex?\line          (c) They both average the same score?}{\lang1033\langfe1033\langnp1033 
\tab 
\par }\pard \ql \li720\ri0\sa120\sl-320\slmult0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin720\itap0 {\lang1033\langfe1033\langnp1033 Give reasons for your answer.
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Play the game with 10 trials for yourself a few times and record your results.
\par {\listtext\pard\plain\hich\af0\dbch\af0\loch\f0 (3)\tab}Collect Data from the }{\b\lang1033\langfe1033\langnp1033 Intro }{\lang1033\langfe1033\langnp1033 page of the Alex and Beth spreadsheet, where you can simulate 20 trials of this game. Also try the }{
\b\lang1033\langfe1033\langnp1033 100 trials}{\lang1033\langfe1033\langnp1033  page.
\par {\listtext\pard\plain\hich\af0\dbch\af0\loch\f0 (4)\tab}Comment on the results of your experiments in parts (2) and (3) and how they relate to your initial idea in (1).
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\par {\listtext\pard\plain\lang2057\langfe1033\langnp2057 \hich\af0\dbch\af0\loch\f0 (6)\tab}A table for the probability distribution of Alex\rquote s winnings after one trial is;
\par }\trowd \trqc\trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \trftsWidth1\trautofit1\trpaddl108\trpaddr108\trpaddfl3\trpaddfr3 \clvertalt
\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1548 \cellx1440\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 
\cltxlrtb\clftsWidth3\clwWidth1680 \cellx3120\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1920 \cellx5040\pard \qc \li0\ri0\sa120\sl-320\slmult0
\widctlpar\intbl\aspalpha\aspnum\faauto\adjustright\rin0\lin0 {a\cell 1\cell 3\cell }\pard \ql \li0\ri0\widctlpar\intbl\aspalpha\aspnum\faauto\adjustright\rin0\lin0 {\trowd \trqc\trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb
\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \trftsWidth1\trautofit1\trpaddl108\trpaddr108\trpaddfl3\trpaddfr3 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 
\cltxlrtb\clftsWidth3\clwWidth1548 \cellx1440\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1680 \cellx3120\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 
\clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1920 \cellx5040\row }\trowd \trqc\trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv
\brdrs\brdrw10 \trftsWidth1\trautofit1\trpaddl108\trpaddr108\trpaddfl3\trpaddfr3 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1548 \cellx1440\clvertalt\clbrdrt
\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1680 \cellx3120\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 
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\trftsWidth1\trautofit1\trpaddl108\trpaddr108\trpaddfl3\trpaddfr3 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1548 \cellx1440\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl
\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1680 \cellx3120\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1920 \cellx5040
\row }\pard \ql \li360\ri0\sa120\sl-320\slmult0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin360\itap0 {Show that the probability distribution of Alex\rquote s winnings after two trials is;
\par }\trowd \trqc\trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \trftsWidth1\trautofit1\trpaddl108\trpaddr108\trpaddfl3\trpaddfr3 \clvertalt
\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1548 \cellx1440\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 
\cltxlrtb\clftsWidth3\clwWidth1680 \cellx3120\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1920 \cellx5040\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 
\clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1920 \cellx6960\pard \qc \li0\ri0\sa120\sl-320\slmult0\widctlpar\intbl\aspalpha\aspnum\faauto\adjustright\rin0\lin0 {a\cell 1\cell 3\cell 9\cell }\pard 
\ql \li0\ri0\widctlpar\intbl\aspalpha\aspnum\faauto\adjustright\rin0\lin0 {\trowd \trqc\trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 
\trftsWidth1\trautofit1\trpaddl108\trpaddr108\trpaddfl3\trpaddfr3 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1548 \cellx1440\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl
\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1680 \cellx3120\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1920 \cellx5040
\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb\clftsWidth3\clwWidth1920 \cellx6960\row }\trowd \trqc\trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 
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 in this coin game. What other measure of central tendency would be more appropriate? Why?
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 trials with coin [1,3]. Show that 
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\par }\pard \ql \li0\ri0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin0\itap0 {
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\lang1033\langfe1033\langnp1033  i.e. a game in which both players can expect to have very similar scores in the long run. One method is to change Alex\rquote s coin from [1,3] to [1,}{\i\lang1033\langfe1033\langnp1033  m}{\lang1033\langfe1033\langnp1033 
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\picscalex100\picscaley100\piccropl0\piccropr0\piccropt0\piccropb0\picw1305\pich600\picwgoal740\pichgoal340\wmetafile8\bliptag290314575\blipupi2304{\*\blipuid 114dd94f281933e0a28e44d584e7b847}
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000041000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02000000000102022253797374656d0000040000002d01010004000000f001000003000000000000}}}}}{where Alex has coin }{\lang1033\langfe1033\langnp1033 [1,}{
\i\lang1033\langfe1033\langnp1033 m}{\lang1033\langfe1033\langnp1033 ], adapt your spreadsheet in part (11) to help improve the fairness of the game by changing the value of }{\i\lang1033\langfe1033\langnp1033 m}{\lang1033\langfe1033\langnp1033 
. Test your value of }{\i\lang1033\langfe1033\langnp1033 m }{\lang1033\langfe1033\langnp1033 on the }{\b\lang1033\langfe1033\langnp1033 100 trials + [1,m]}{\lang1033\langfe1033\langnp1033  page of the }{\field\fldedit{\*\fldinst {
\lang1033\langfe1033\langnp1033  HYPERLINK "http://www.planetqhe.com/frames/comp2frames.html" }{\lang1033\langfe1033\langnp1033 {\*\datafield 
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\lang1033\langfe1033\langnp1033  Alex and Beth spreadsheet. }}}{\lang1033\langfe1033\langnp1033 
\par (13) Choose one of these methods to improve the fairness of the game.  
\par {\listtext\pard\plain\hich\af0\dbch\af0\loch\f0 (a)\tab}}\pard \ql \fi120\li720\ri0\sa120\sl-320\slmult0\widctlpar\jclisttab\tx720\aspalpha\aspnum\faauto\ls7\adjustright\rin0\lin720\itap0 {\lang1033\langfe1033\langnp1033 Fix Alex\rquote 
s coin at [1,3] but change the probability of his throwing a 3.
\par {\listtext\pard\plain\hich\af0\dbch\af0\loch\f0 (b)\tab}Fix Alex\rquote s coin at [1,3] but change his starting value.
\par }\pard \ql \li360\ri0\sa120\sl-320\slmult0\widctlpar\aspalpha\aspnum\faauto\adjustright\rin0\lin360\itap0 {\lang1033\langfe1033\langnp1033 
Include in your analysis a comparison of the use of the mean and the median as measures of central tendency. You can test your fair game model on the }{\b\lang1033\langfe1033\langnp1033 100 trials + prob }{\lang1033\langfe1033\langnp1033 and }{
\b\lang1033\langfe1033\langnp1033 100 trials + start}{\lang1033\langfe1033\langnp1033  pages of the }{\field\fldedit{\*\fldinst {\lang1033\langfe1033\langnp1033  HYPERLINK "http://www.planetqhe.com/frames/comp2frames.html" }{
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\lang1033\langfe1033\langnp1033  Alex and Beth spreadsheet. }}}}}{
\par }}