There are 38 total spaces on the wheel

• 18 black spaces
• 18 red spaces
• 2 green spaces

Each green space is either marked 0 and 00. Each red and black space is labeled with a number between 1 and 36.

The probability of landing on a black, red, or green space is 18/38, 18/38 and 2/38 respectively. The probability on landing on one specific number is 1/38.

The formula for winning when betting on a specific color:

\[ \begin{equation} \tiny P(\text{red or black}) \times P(\text{bet}) + P(\text{not red or black}) \times P(-\text{bet}) = \left(\frac{18}{38}\right) \times (1) + \left(\frac{20}{38}\right) \times (-1) = -0.053 \end{equation} \] \[ \begin{equation} \tiny P(\text{green}) \times P(\text{bet}) + P(\text{not green}) \times P(-\text{bet}) = \left(\frac{2}{38}\right) \times (1) + \left(\frac{36}{38}\right) \times (-1) = -0.895 \end{equation} \] The formula for winning when betting on a specific number:

\[ \begin{equation} \tiny P(\text{number}) \times P(\text{bet}) + P(\text{not number}) \times P(-\text{bet}) = \left(\frac{1}{38}\right) \times (10) + \left(\frac{37}{38}\right) \times (-10) = -9.737 \end{equation} \]

Jack is at a casino and decides to play a few games of roulette. He currently has 100 dollars. He decides to bet 1 dollar on the first round and if he wins, then he’ll bet his winnings on the next round. But if he loses, then he’s back to betting 1 dollar. Jack does this for each of the 100 rounds.

Jack is a very superstitious guy. Jack’s lucky color is black and his lucky number is 7. He decides to only bet on these two outcomes. He wants to know the probability of winning if he bets on black for the first 50 games and the probability of him winning if he bets on 7 for the last 50 games. If he wins betting on black, then his bet gets doubled. If he wins betting on 7, then his bet gets multiplied by 10.

\[ \begin{equation} \tiny P(\text{black}) \times P(\text{bet}) + P(\text{not black}) \times P(-\text{bet}) = \left(\frac{18}{38}\right) \times (1) + \left(\frac{20}{38}\right) \times (-1) = -0.053 \end{equation} \]

From the graph on the previous slide, we can see that the more Jack bets on black, the higher Jack’s chances are of the roulette ball landing on black. Each time that the graph goes up, that means that Jack won that round. But every time the graph stays flat, that means that Jack lost the round.

Based on the graph from the previous slide, Jack’s winnings fluctuate quite a bit. If we compare it to the graph that plots his probability of landing on black, the graph of his winnings goes down when his probability of landing on black is flat. Then, of course, his graph of winnings goes up when his probability of landing on black goes up.

\[ \begin{equation} \tiny P(\text{7}) \times P(\text{bet}) + P(\text{not 7}) \times P(-\text{bet}) = \left(\frac{1}{38}\right) \times (10) + \left(\frac{37}{38}\right) \times (-10) = -9.737 \end{equation} \]

According to the graph, Jack’s chances of winning are significantly lower betting on 7 than betting on black. This is mirrored in the probability of landing on black compared to 7. Similar to the graph showing the probability of landing on black, every time the graph increases, that means that Jack won the round. But for every data point that keeps the graph flat, means that Jack lost the round.

Just like the graph that plots Jack’s winnings from landing on black, the graph from the previous slide matches up with the graph that shows Jack’s probability of landing on 7. Every time that the graph of Jack’s winnings of landing on 7 goes down, that is represented by the flat portions from the probability graph, which means that Jack lost those rounds. Every time that the previous graph went up, that represented that Jack won that round.

According to all of the graphs, it seems that Jack will end up walking away from the table with less money than he started with. Maybe Jack’s “lucky” color and number aren’t so lucky. Or maybe he just needs to switch up his bets the next time he plays roulette instead of picking the same bet for 50 rounds straight.

Don’t be like Jack. Don’t bet all of your money on black just because you think it’s “lucky”. Make an algorithm in R that will tell you what to bet on instead ;).