Exploring The Math Behind Roobet's Towers

Disclaimer: Roobet may change their odds for the game at any point, meaning that the results of the calculations are not guaranteed to be exact at the time you’re reading this.

Before finding the house edge that Roobet takes from their Tower’s game, let’s quickly go over how to calculate win probabilities and the expected value of a given game.

Let’s take Roobet’s Easy mode, for example. In this game mode, there are 4 tiles per level. 3 of the tiles will allow you to advance to the next level, while one of the doors has a blank space behind it and will cause you to lose your bet. This means that you have a 75% chance of making it to the next level.

Since the probability of making it past each layer is 75%, we can find the probability of successfully passing X layers by simply multiplying 75% with itself x times:

(.75)^x

Below, we can see a table that shows the probability of passing X layers if we have a 75% chance of passing each one. We can expect to pass all 8 layers just over 10% of the time.

So, now that we know the probability of passing each number of layers, we can calculate the expected value by incorporating the payout into the equation.

To calculate the expected value of a given bet, we can use the following formula:

Expected Value = (Probability of Winning x Profit) – (Probability of Losing x Bet Amount)

Let’s look at beating 8 layers on Easy mode (75% success rate per floor) as an example. The payout for passing 8 layers on Easy mode is $7.21 on a $1 bet, which means that you’d profit $6.21 since your initial $1 stake is included in the payout. Let’s plug this information into our expected value formula:

Expected Value of Passing 8 Layers on Easy Mode = (10.01% x $6.21) – (89.99% x $1) = -$0.278.

We can see that the expected value of this bet is -$0.278 or -27.8% since it’s on a $1 bet. This means that if we were to play this game many many times over, we could expect to lose about $0.278 per game on average. This is a pretty bad return, considering that typical casino games only have about a 1% to 5% house edge.

Now that we have calculated the expected value of the game, let’s look at the expected value for all possible floors for Towers on Easy mode.

From the table below, we can see that not all bets have the same expected value of -27.8%. If you were to only play past the first layer and then cashout, the Expected Value of the bet would be -4%.

We can see an interesting pattern in the expected values by layer. It turns out that the expected value can be calculated as 1 – (96%)^X, where X is the layer number. This means that Roobet is taking a 4% cut each time the player chooses to play another layer in an attempt to move on to the next one, so the house edge compounds over the course of a full game.

When we look at the expected values of all of the other difficulties (Medium, Hard, Expert, and Crazy), we find that Roobet also takes a 4% house edge per level, leading to the same Expected Value by layer. The only pieces that change when changing the difficulty level are the win probability and then payout, but they will still lead to the same expected value.

Easier modes will lead to a more steady bankroll (although negative), while harder modes will lead to a bankroll that is much more variable, while also trending in the negative direction.

Lastly, let’s calculate the probability of beating each of the 5 game difficulties. We’ve already found that there’s about a 10.01% chance of making it past 8 layers on Easy mode, but what about the others?

Using the same calculation from before, we have the following results:

• 3.9% (1 in 25.6) chance of making it past 8 layers on Medium mode;
• 0.391% (1 in 256) chance on Hard mode;
• 0.0152% (1 in 6,561) chance on Expert mode;
• 0.00153% (1 in 65,536) chance on Crazy mode.

Keep in mind that Roobet does have a max win amount on this game, so you have to make sure that your bet amount is low enough to win the max price if you do go for some of these long shot bets.

Overall, we’ve found that Roobet takes a 4% house edge at each layer of their Towers game. The probabilities of making it all the way through the game vary based on the game difficulty, ranging from 1 in 10, to 1 in over 65,000.

We’ve covered how to calculate the probability of winning a given bet, and how to then calculate the expected value of a bet given the win probability and payout.

Thanks for reading, and I hope you’ve learned some useful information on calculating risks and rewards in games of chance!

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