Probabilities are omnipresent in our day-to-day lives. However, most daily situations have dependencies, making the likelihood of events hard to estimate. Games of chance are different. Not only are the rules explicit, allowing for a clear measurement of likelihoods, they are also repeatable, making them a great case study for anyone interested in probabilities.
A game of chance you might be familiar with is roulette, which we will be using to display the power of the Fibonacci sequence in comparison to the Martingale. For the purpose of this case study, we will be working with the French roulette as it has better odds at every bar than the American one — 37 vs. 38 divisions. The double zero on the American roulette wheel creates an additional bar where the ball may fall, lowering odds for every number. This has an especially negative impact when placing outside bets such as black or red, odd or even, and high or low. In the case of French roulette, the zero lowers the odds on outside bets to 48.65%, bringing the casino edge to 2.7%. The additional American double zero pushes odds on outside bets to 47.37%, helping the casino to a 5.26% edge. Sooner or later, with enough spins, the House will surely rake in more money than it pays out to players.
One way players can compensate for the House edge is to throw good money after bad money — gamblers increase their bets to recuperate their lost money and to profit. However, there are downsides to this strategy. The House sets limits on outside bets, meaning that you cannot constantly increase your bets. These limits vary from casino to casino, with common minimum and maximum limits ranging from $25 to $2,000. Once a player hits the upper threshold, he can no longer increase his bets, and can only count on luck to turn a profit. On the opposite side of the spectrum, a player cannot start extremely small to increase his buffer before reaching the upper limit.
In a strategy like the Martingale where the player doubles his bet at every loss, he would pass the table limit after seven consecutive losing spins. On top of the capital required to double bets, even when starting with the lower limit of $25, the seventh spin requires $1,600 in addition to the $1,575 lost in all previous six spins.
Considering the capital requirements, the return on spins (ROS), is not always rewarding. On the first spin, the ROS will be 100% if the player wins. However, on the second spin, as we factor in the previous loss, the ROS falls to 33.33%. On the seventh spin, the potential return plunges to 0.79%.
On the upside, the probability of losing seven spins in a row is statistically small. Remembering that every spin is independent of all others, the probability of losing the first one is 51.35% or (19/37)^1 and losing seven spins in a row is 0.94% or (19/37)^7.
To get a better feel for these odds, the following python script simulates a thousand runs where the gambler takes ten spins:
The outcome of the 1,000 simulations shows low returns with a heavy downside of potential losses: 50% of visits would result in winnings of $100 or more, yet at a cost of leaving on average $36.63 at the casino. Materially, losing seven consecutive spins might not only occur more often than the gambler would hope, it would also have an atrocious impact on his wallet.
To look at the problem from another side, the following python script simulates the number of spins it would take a gambler to leave the table with $100:
The median number of spins required using the Martingale strategy ends up being double the amount of the best-case scenario. Moreover, the distribution is heavily right-skewed — the simulations are truncated at 100 runs to prevent neverending losing streaks — which indicates that recovering from a losing streak is not always possible.
Now, the Martingale might make you wonder why any rational player would ever choose to play roulette, other than the adrenaline rush that comes with watching a ball spin around the wheel. And though that is probably the only reason why one should play the game, there are other strategies that can work better for a player.
One such strategy, that is less aggressive on a wallet and works better with casino limits, is the Fibonacci. The Fibonacci strategy consists of using the numbers from the Fibonacci sequence to determine the bets to place. The sequence is easy to remember as it involves summing the two previous numbers, starting with 0 and 1:
Hence, the beginning of the sequence follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …
With every loss, the player moves up the sequence for his next bet. On a win, he moves back two numbers in the sequence. In theory, Fibonacci allows players to leave the table with money even though they may have lost more games than they have won.
The following python script simulates a thousand runs where the gambler takes ten spins using the Fibonacci sequence:
From the outcome of these simulations, we see that the maximum loss from this strategy is much smaller, and in turn, the mean is higher. Even so, all three quartiles are lower than with the Martingale.
The more conservative approach of Fibonacci is also visible when calculating the number of spins required for a player to leave the table with $100, which the following python code simulates:
The quartiles shift right which implies a gambler will need to sit through more spins before he can go home to a chicken dinner!
Altogether, the casino is expected to make money on roulette players. However, if you do decide to play, understanding these strategies can make the experience that much more adrenaline-filled. And if you prefer keeping your money in your pocket, you can just re-run the python code!
Note: This article is not gambling advice, simply a display of one of the uses of the Fibonacci sequence.