Reddit Post link: https://www.reddit.com/r/CompetitiveTFT/comments/1884ab9/tft_math_tome_of_traits/
Hi everyone, I’m Gouphee and I've barely been able to play TFT recently. However, I do randomly think about math, and as a result, I randomly thought about probabilities for tome of traits when I couldn’t sleep one night. I did some napkin math and then wrote a python program and found out some results that I personally didn’t expect. There have been many posts on reddit about specific odds for a specific set, as well as what units are best for tailoring your tome, but I was interested in generalizing the math behind tome of traits odds and seeing if there were any interesting conclusions to draw.
Before I go into the results, I want to lay the groundwork and assumptions. Currently, Tome of traits works with the following specific breakpoints.
| Number of traits on board | Number of tailored emblems |
|---|---|
| 0-5 | 0 |
| 6-7 | 1 |
| 8-9 | 2 |
| 10-11 | 3 |
| 12+ | 4 |
There are some useful tips when making your boards in such a way. The first one is that when you are in a given “tier” of number of traits on board, you want to have the minimum. This number is usually even, so a good heuristic is never have an odd number of traits in before popping your tome. For example, you would rather have 6 than 7 traits. This is because when you have your tailored choices, 1/6 would be a higher chance than 1/7. While this may be obvious, I think there are plenty of people who may not try and reduce the number of traits and instead only focus on increasing the number of traits, and I haven’t really seen it discussed much (granted I didn’t explicitly look hard to find out if it had been).
The second is that if you have a trait in your board that doesn’t have an emblem associated with it, you effectively reduce the number of choices for your tailored emblem. As an example, in Set 10, EDM is a non-emblem trait. If it was one of your 6 traits, you would have a 1/5 chance instead of a 1/6 chance for your tailored choice.
Like I said before, many posts have talked about these sorts of tips. I will also be making a post that shows what this sets best units for tomes are as well, although it may be obvious to most already (hint: EDM, Akali, legendaries, and superfans).
One other important assumption that I made is that the tailored emblems are rolled first before the non-tailored ones. I have not seen any information on whether or not this is true, so I would need confirmation. The math slightly changes if non-tailored emblems are first, or if each spot is randomly decided as non-tailored/tailored. If I am incorrect, I will update the model accordingly, although I hope this is not the case as it makes it much more complicated than it already is.
Although I said I was interested in generalized math for tomes, I will be using set 10 as an example simply to showcase actual numbers instead of arbitrary variable letters. Now that we have established basic assumptions, we can go into the actual math aspect of tomes.
Tomes are essentially a combination of 4 emblems without replacement. As a result, if you want to calculate the chance that you roll a specific trait in your tome of traits, you would calculate the chance that you don’t get the specific trait on each individual roll, multiply them, and then subtract from 1. Before we work out the generalized equation, I find it easiest to understand some of these probability problems by working through an example.
Set 10 has 23 traits that are available as emblems. Let’s suppose that we have 8 traits on our board, and we are looking for exactly a Heartsteel emblem. When we pop the tome, we roll an emblem 4 times, without replacement.
We have two tailored choices first. The odds that we don’t hit Heartsteel are 7/8 for the first one, and 6/7 for the second one. The second choice has different odds because there is one less choice in the possible pool, since it was selected in our first roll.
For the two non-tailored choices, we have 21 emblems still available since 2 have been selected. As a result, the odds that we don’t hit are 20/21 and 19/20 for the third and fourth rolls respectively. The overall odds that we don’t hit is the product of all of these rolls, which simplifies down to 19/28. To calculate the odds that we do hit, we subtract the odds that we don’t from 1. Thus, the odds we do hit are 9/28 which is about 32.5%.
Hopefully that made sense. You can use this same concept for a variety of values. Our next goal is to generalize this to an equation with variable values. To get to our equation, we have to do a little math trickery. You might notice that there’s two distinct groups for the fractions. The first group is the tailored rolls and the second group is the non-tailored rolls. When we group the fractions like so, you might see a pattern.
The numbers go down by one in each group. Furthermore, the numerator of the previous probability roll is the denominator of the next probability roll in each group. That means, to simplify, you would just take the numerator of the last roll in each group and the denominator of the first roll of each group.