At the Crown Casino in Melbourne, Australia, some roulette wheels have 18
slots coloured red, 18 slots coloured black, and 1 slot (numbered 0) coloured
green. The red and black slots are also numbered from 1 to 36. (Note that
some of the roulette wheels also have a double zero, also coloured green, which
nearly doubles the house percentage.)
You can play various ‘games’ or ‘systems’ in roulette. Four possible games are:
• A. Betting on Red
This game involves just one bet. You bet $1 on red. If the ball lands on red
you win $1, otherwise you lose.
• B. Betting on a Number
This game involves just one bet. You bet $1 on a particular number, say
17; if the ball lands on that number you win $35, otherwise you lose.
• C. Martingale System
In this game you start by betting $1 on red. If you lose, you double your
previous bet; if you win, you bet $1 again. You continue to play until you
have won $10, or the bet exceeds $100.
• D. Labouchere System
In this game you start with the list of numbers (1, 2, 3, 4). You bet the sum
of the first and last numbers on red (initially $5). If you win you delete the
first and last numbers from the list (so if you win your first bet it becomes
(2,3)), otherwise you add the sum to the end of your list (so if you lose
your first bet it becomes (1, 2, 3, 4, 5)). You repeat this process until your
list is empty, or the bet exceeds $100. If only one number is left on the list,
you bet that number.
Different games offer different playing experiences; for example, 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. The aim of this assignment is to compare
the four games above using the following criteria:
1. The expected winnings per game;
2. The proportion of games you win;
3. The expected playing time per game, measured by the number of bets
made;
4. The maximum amount you can lose;
5. The maximum amount you can win.
ROULETTE 543
24.3.1 Simulation
For each game write a function (with no inputs) that plays the game once
and returns a vector of length two consisting of the amount won/lost and how
many bets were made. Then write a program that estimates 1, 2, and 3, by
simulating 100,000 repetitions of each game. Note that a game is won if you
make money and lost if you lose money.
24.3.2 Verification
For games A and B, check your estimates for 1 and 2 by calculating the
exact answers. What is the percentage error in your estimates for 100,000
repetitions?
For each game, work out the exact answers for 4 and 5. Of course, if this is not
close to the answer given by your simulation, then you should suspect that
either your calculation or your program is erroneous.
24.3.3 Variation
Repeat the simulation experiment of Part 24.3.1 five times. Report the minimum and maximum values for 1, 2, and 3 in a table as follows:
Exp. winnings Prop. wins Exp. play time
Game min–max min–max min–max
A
B
C
D
Modify your program from Part 24.3.1 so that in addition to estimating the
expected winnings, expected proportion of wins, and expected playing time,
it also estimates the standard deviation of each of these values. (You may use
the built-in function sd(x) to do this.) For a single run, consisting of 100,000
repetitions of each game, report your results in a table as follows:
544 STUDENT PROJECTS
Winnings Prop. wins Play time
Game mean, std dev mean, std dev mean, std dev
A
B
C
D
For which game is the amount won most variable?
For which game is the expected playing time most variable
I can give screenshots if that is hard to read
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