## Visualizing Probability: Roulette

I wrote a post over on the Society of Information Risk Analysts blog and I was having so much fun, I just had to continue. I focused this work on the American version of Roulette, which has “0” and “00” (European version only has “0” producing odds less in favor of the house). The American versions also have “Five Numbers” option to bet, which the European version doesn’t have.

According to this site, the American version of roulette could do about 60-100 spins in an hour, I figured maybe 4 hours in the casino and being conservative, I decided to model 250 iterations of roulette. I then chose $5 bets, which isn’t significant, changing the bet would only change the scale on the left, not the visuals produced. I then ran 20,000 simulations of 250 roulette spins and recorded the loss or gains from the bets along the way. One way to think of this is like watching 20,000 people play 250 spins of roulette and recording and plotting the outcomes.

I present this as a way to understand the probabilities of the different betting options in roulette. I leveraged the names and payout information from fastodds.com. The main graphic represents the progression of the 20,000 players through the spins. Everyone starts at zero and either goes up or down depending on lady luck. The distribution at the end shows the relative frequency of the outcomes.

Enough talking, let’s get to the pictures.

### Betting on a single number

What’s interesting is the patterns forming from the slow steady march of losing money punctuated by large (35 to 1) wins. Notice there would be a few unlucky runs with no wins at all (the red line starts at zero and proceeds straight down to 1250). Also notice in the distribution on the right that just over half of the distribution occurs under zero (the horizontal line). The benefit will always go to the house.

### Betting on a pair of numbers

Same type of pattern, but we see the scale changing, the highs aren’t as high and the lows aren’t as low. None of the 20,000 simulations lost the whole time.

### Betting on Three Numbers

### Betting on Four Number Squares

### Betting on Five Numbers

### Betting on Six Numbers

### Betting on Dozen Numbers or a Column

### Betting on Even/Odd, Red/Green, High/Low

And by this point, when we have 1 to 1 payout odds, the pattern is gone along with the extreme highs and lows.

### Mixing it Up

Because it is possible to simulate most any pattern of betting, I decided to try random betting. During any individual round, the bet would be on any one of the eight possible bets, all for $5. The output isn’t really that surprising.

### Rolling it up into one

Pun intended. While these graphics help us understand the individual strategy, it doesn’t really help us compare between them. In order to do that I created a violin plot (the red line represents the mean across the strategies).

Looking at the red line, they all have about the same mean with the exception of Five Numbers (6-1). Meaning overtime, the gambler should average to just over a 5% loss (or a 7% loss with five number bets). We can see that larger odds stretch the range out, which smaller odds cluster much more around a slight loss. The “scatter” strategy does not improve the outcome and is just a combination of the other distributions. As mentioned, the 6-1 odds (Five Numbers) bet does stick out here as a slightly worse bet than the others.

Lastly, I want to turn back to a comment on the fastodds.com website:

While I may disagree that the only bets to avoid are limited to those (had to get that in), I also disagree with the blanket statement. Since they all lose more often than they win, trying to get less-sucky-odds seems a bit, well, counter-intuitive. I would argue that the bets to avoid are not the same for every gambler. The bets should align with the tolerance of the gambler. For example, if someone is risk-averse, staying with the 2-1 or 1-1 payouts would limit the exposure to loss, while those more risk-seeking, may go for the 17-1 or 35-1 payout – the bigger the risk, the bigger the reward. Another thing to consider is that the smaller odds win more often. If the thrill of winning is important, perhaps staying away from the bigger odds is a good strategy.

Now that you’re armed with this information, if you still have questions, the Roulette Guru is available to advise based on his years of experience.