1 Roulette [22 marks]In Crown Casino a roulette wheel has 18 slots coloured red,

1 Roulette [22 marks]In Crown Casino a roulette wheel has 18 slots coloured red, 18 slots coloured black and 1 slot (0) coloured green. The red and black slots are also numbered from 1-36. You can play various “games” or “systems” in roulette. Four possible games are: A. Betting on odd number
This game involves just one bed. You bet $1 on odd number. If the ball lands on odd number you win $1 (and get your bet back), otherwise you lose $1. B. Betting on two numbers
This game involves just one bet. You bet on a list of two numbers say (6, 17); if the ball lands on that one of those two numbers you win $17 ( and get your bet back), otherwise you lose $1. C. Martingale System
In this game you start by betting $1 on odd number. If you lose, you double your previous bet; if you win, you bet $1 again. You continue to play until you have won $20, or the bet exceeds $200. D. Labouchere System
In this game you start with the list of numbers (1, 2, 3, 4, 5). You bet the sum of the fifirst and last numbers on odd number (initially $6). If you win you delete the fifirst and last numbers from the list, which becomes (2,3,4), otherwise you add the sum to the end of your list (which becomes (1, 2, 3, 4, 5, 6)). You repeat this process till your list is empty, or the bet exceeds $200. (If only one numbers are left on the list you bet that that number). Difffferent games offffer difffferent playing experiences e.g. some allow you to win more often than you lose, some let you play longer, some cost more to play and some risk greater losses. Our aim is to compare the four game types above using: a. the expected winnings/game b. the proportion of games you win c. the expected playing time (measured by the number of bets) d. the maximum amount you can lose e. the maximum amount you can win 1i. Write a MATLAB program which estimates (a), (b) and (c) for Games A-D by simulating 100,000 repe- titions. The program should play a difffferent game, depending whether the input argument is ’A’ or ’B’ etc. The fifirst line should be function roulette(gametype). There should be one subfunction for each type of game which simulates a single game of that type. ii. Check your program estimates for (a) and (b) for Game A by calculating the exact answers. What is the percentage error in your estimates for 100,000 repetitions, as measured by the mean of 5 runs, compared to the exact result? iii. Using mathematical arguments, calculate the exact answers for (d) and (e) for Games A-D. (Of course you might want to use your program to investigate this problem to help you fifind the theoretical answer). iv. Summarise your results in a table as follows: Game Exp. winnings Prop wins Exp. play time Max loss Max win ABCDIn each entry of the fifirst three columns of the table, enter the range of values (minimum, maximum) from 5 separate runs, each of 100,000 repetitions. Compare the variability of expected winnings in Games A and C, as measured by the half-range of 5 runs i.e. (max-min)/2. Compare the variability of expected winnings and expected playing time in Game D.
Requirements: 1111   |   .doc file