The International School of Toulouse Department of Mathematics. 
Math Higher Portfolio assignment – Modelling Coins
TYPE III	

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.

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. 
Each player begins with a starting value of  1.
They multiply the starting value by the number they throw on their respective coins.

Section One; The original game;
(1)	After many games, do you expect that the outcome will be: 
         (a) Alex gets a higher score than Beth?
         (b) Beth gets a higher score than Alex?
         (c) They both average the same score?	
Give reasons for your answer.
(2)	Play the game with 10 trials for yourself a few times and record your results.
(3)	Collect Data from the Intro page of the Alex and Beth spreadsheet, where you can simulate 20 trials of this game. Also try the 100 trials page.
(4)	Comment on the results of your experiments in parts (2) and (3) and how they relate to your initial idea in (1).
Section Two; Comparing the Expectations;
(5)	Suppose that Alex and Beth pay the game for n trials. Let the expected value of Beth’s winnings after n trials. Show that 
(6)	A table for the probability distribution of Alex’s winnings after one trial is;
a	1	3	
	½	½	
Show that the probability distribution of Alex’s winnings after two trials is;
a	1	3	9	
	¼	2/4	1/4	
 
And show that the probability distribution of Alex’s winnings after three trials is;
a	1	3	9	27	
	1/8	3/8	3/8	1/8	

(7)	Let denote the expected value of Alex’s winnings after n trials. Prove using the binomial theorem or otherwise that .
(8)	Compare, for n trials, the shape of the distributions of and  by using a spreadsheet. Hence comment on the limitations of the application of the formula  as a measure of the expected winner in this coin game. What other measure of central tendency would be more appropriate? Why?
Section Two; Modelling the expected winner;
(9)	 Let denote the median value of Beth’s winnings after n trials. Show that 
(10) Let denote the median value of Alex’s winnings after n trials with coin [1,3]. Show that 

(11) Use a spreadsheet to compare the values of  and for n =1...20.

Section Three; Making the game fair;
(12) There are various approaches to making the game fair - 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’s coin from [1,3] to [1, m] and vary the value of m. By deriving a general formula for where Alex has coin [1,m], adapt your spreadsheet in part (11) to help improve the fairness of the game by changing the value of m. Test your value of m on the 100 trials + [1,m] page of the  Alex and Beth spreadsheet. 
(13) Choose one of these methods to improve the fairness of the game.  
(a)	Fix Alex’s coin at [1,3] but change the probability of his throwing a 3.
(b)	Fix Alex’s coin at [1,3] but change his starting value.
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 100 trials + prob and 100 trials + start pages of the  Alex and Beth spreadsheet.