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I have sought the best possible strategy for playing roulette at the casino I am at.  I consider the best possible strategy one that has the median above the stake amount and there does not exist an alternative strategy that has higher Q1 or Q3 value.  It is not guaranteed that a best possible strategy exists.  The strategies considered are different 25 different multipliers between 1.1 and 2, where the bet is the ceiling of the multiplication. To test these strategies, I ran 10,000 simulations of each configuration. The slide show of plots describes the outcomes by strategy for each of $500, $1000 and $2000 stakes. I’m not willing to risk more than $2000 at this, so I didn’t get into analyzing details of higher stakes. The maximum bets available make higher returns not likely with higher stakes anyway. The median outcome of the $500 stake never gets to a positive return. The $1000 stake does have medians of positive return, but the $2000 stake has generally higher Q1 values. These plots do not show the entire picture because the table to play is one of the decisions to make and is not visible.  The next plot shows the difference between the games for strategies in the range 1.625 and 1.775.  I didn’t choose the 1.85 strategy because I am risk averse (don’t laugh) and running separate simulations showed that the results in the previous plot were a fluke and have Q1 values that follow the trend of the other strategies.

The criteria for a best possible strategy is satisfied with $25 minimum bet and limiting play to 50 games.  I used this data to chose a betting progression that approximates the four strategies in the range displayed and uses no single dollar chips (for easier betting- also good to consider when analyzing decisions). I then performed another simulation with these choices to check the outcomes, which are displayed below.

You can see that the expected value of playing the strategy with $2000 is $1842.59 and the median outcome is a positive return.  While over half of the outcomes seen have positive value, the expected value is a loss of $147.41.  This is because about 1/4 of the games lose $400 to $1400, while the winnings only get to about $750.  This strategy I developed is a losing strategy because I can’t expect to make money in the long run.  I also can’t expect to make any money on any particular game without violating all my training as a mathematician.  So, I am going to play expecting that I will lose $147.41.