## Sunday, 27 July 2014

### Monte Carlo Simulation of Western 649

Someone recently expressed disbelief to me that Western 649 is actually a smarter gamble than a typical 50/50 draw, so I decided to run a short Monte Carlo simulation to prove that my previous analysis based on probability theory was correct. I used Excel to generate 500,000 random numbers between 1 and 13,983,816 to simulate one Western 649 draw with 500,000 random combinations played. The criteria for determining which of my simulated numbers were winners were as follows:

A total of 3,000 draws (each with 500,000 random combinations played) were simulated. The tables below summarize the results of my simulation:

 Number of winners in the simulated Western 649 draws

 Summary of WCLC's profit and house edge, and Western 649 expected value

In the first table, we see that the expected values generally agree well with the averages from the simulation, apart from there being a few more jackpot winners than expected from probability theory. At most I had three winners sharing the \$50,000 prize, which from my previous analysis I calculated had about 1 in 753 chance of occurrence in a draw with 500,000 random combinations. I also had two winners sharing the \$1,000,000 prize in one of my simulated draws, which I calculated previously to have about 1 in 1,621 chance of occurrence.

The second table shows that the worst individual draw for the WCLC resulted in a net loss of \$920,090. But these losses were generally few and far between. The best draw for the WCLC produced \$139,330 in profit. After 3,000 simulated draws, the WCLC earned a total of more than \$251 million on the Western 649 game, averaging nearly \$84,000 in profits per draw. Note that these numbers ignore operating costs to run the lottery: in real life, the WCLC has to pay for equipment and personnel to run the lottery, sell tickets, etc. However, it's still illustrative of how lucrative running this lottery is (WCLC runs two Western 649 draws every week). The simulation results showed an overall house edge of 33.50%. This corresponds to players losing, on average, \$0.335 of every \$1 spent playing Western 649. This is only about 3.4% off what the analysis based on probability theory indicated. The expected average profit was \$86,665.97 per draw, which corresponds to 34.67% house edge and players losing \$0.3467 of every \$1 wagered.

Since this is a random number simulation with a finite number of trials, minor differences between the simulation results and the results from probability theory should be expected. The average would approach the expected value as the number of trials approaches infinity. I've plotted the overall house edge against the number of simulated draws in the graph below to show that my simulation results do tend to approach the expected value.

 Overall house edge versus number of simulated draws

In summary, a simple Monte Carlo simulation of 3,000 Western 649 draws supports the results of my previous analysis based on probability theory. The expected value of Western 649 is better than the expected value of a typical 50/50 draw.