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Assignment on Stratosphere Casino in Las Vegas

## Comparing Four Roulette Games

The project can be completed using Excel

At the Stratosphere Casino in Las Vegas, 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.

You can play various `games' or `systems' in roulette. Four possible games are:

g1) Betting on Red. This games involves just one bet. You bet 1\$ on red. If the ball lands on red you win 1\$ (that is you get back 2\$: the one you bet and the one you win), otherwise you lose.

g2) Betting on a Number. This game involves only one bet. You bet 1\$ on a particular number, say 23; if the ball lands on that number you win 35\$, otherwise you lose.

g3) Martingale System. This game involves more than one bet. 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\$.

g4) Labouchere System. This game involves more than one bet. In this game you start with the list of number (1, 2, 3, 4). You bet the sum of the First and last numbers on red (initially, thus, 5\$). If you win you delete the First and the last numbers from the list (so if you win your First bet, the list becomes (2, 3)), otherwise you add the sum to the end of your list (so if you lose your First bet, the list 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 by using the following criteria:

c1. The expected winnings per game; c2. The proportion of games you win (note that a game is won if you make money and lost if you lose money); c3. The expected playing time per game, measured by the number of bets made; c4. The maximum amount you can lose; c5. The maximum amount you can win.

1. Â For each game run a study that estimates c1, c2, c3, c4 and c5 by simulating 10 000 repetitions of the game. Compare the values you obtain.

2. Â For games g1 and g2, check your estimates for c1 and c2 by calculating the exact answers. What is the percentage error in your estimates for 10 000 repetitions?

3. Â As for c1 and c2, plot running average together running variance for the ve games. Based on visual investigation of such plots, do you think the estimates you get are reliable? If not, try to increase the number of repetitions of your study.

4. Â Modify the study you ran for point a) so that, in addition to estimating the expected winning, expected proportion of wins, and expected playing time, it also estimates the variance of each of these values. For which game is the amount won most variable? For which game is the expected playing time most variable?

A deck of 100 cards - numbered 1, 2, . . . , 100 - is shufflÂ­ed and then turned over one card at a time. We say that a hit occurs whenever card i is the ith card to be turned over, i = 1, . . . , 100. Simulate 10 000 repetitions of the game to estimate the expectation and variance of the total number of hits