TFT-Rolling-Odds-Calculator

A calculator that provides the odds of hitting your units in TFT, made with very simple HTML and Chart.js

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Update as of 31/7/24: Set 12 bag sizes and shop odds. 30/25/18/10/9 for each cost, 14/13/13/12/8 distinct champions for each cost. Also included toggle for comparison between sets.

Update as of 29/5/24: Set 11 bag sizes and shop odds. 22/20/17/10/9 for each cost, 13/13/13/12/8 distinct champions for each cost. Updated level 7/8 cost probabilities

Update as of 26/11/23: Set 10 bag sizes and shop odds. 22/20/17/10/9 for each cost, 13/13/13/13/8 distinct champions for each cost

Update as of 26/8/23. Works as long as there are 29 1-costs, 22 2-costs, 18 3-costs, 12 4-costs, 10 5-costs in the pool, and number of distinct champions of each cost are 13, 13, 13, 12, 8 respectively.

Brief explanation of the math

In TFT, a roll gives you 5 shops. Each shop rolls for the cost of the unit first (based on your current level), followed by the unit itself (based on the number of units left & the pool size for units of that cost). So for example if we are rolling for Yuumi,

$$ P(\text{hit Yuumi in this shop}) = P(\text{roll 2-cost})\cdot\frac{\text{number of yuumis left}}{\text{number of 2-costs left}} $$

Brute-force probability calculation is cumbersome as $\text{number of yuumis left}$, etc. may decrease as you roll. Thus, rolls can be modelled with Markov Chains.

Suppose we want to find the probability of finding x yuumis after exactly 2 shops. Below is an example of a Markov Chain given some number of yuumis and 2-costs out of the pool already:

We start with 0 yuumis (on the left), and after 2 shops we must be in one of the 3 states above. The arrows represent the transition probabilities, i.e. probability of going from one state to another. We can write the corresponding transition matrix:

$$ \begin{pmatrix} 0.977 & 0.023 & 0 \ 0 & 0.976 & 0.024 \ 0 & 0 & 1 \end{pmatrix} $$

Since we roll twice, we square the transition matrix as follows:

$$ \begin{pmatrix} 0.977 & 0.023 & 0 \ 0 & 0.976 & 0.024 \ 0 & 0 & 1 \end{pmatrix}^2 = \begin{pmatrix} 0.955 & 0.045 & 0.0006 \ 0 & 0.952 & 0.047 \ 0 & 0 & 1 \end{pmatrix} $$

Because of how matrix multiplication works, we can read off "P(start with i yuumis and end with j yuumis)" from the (i, j)-th entry of this matrix! Thus we have a 0.955 chance of getting no yuumis, a 0.045 chance of getting 1 yuumi, and a 0.0006 chance of getting 2 yuumis after seeing 2 shops.

For the purposes of this calculator, the transition matrix $M$ is 10 x 10 (To calculate probabilities of getting up to 9 yuumis), and we basically compute $M^{5g/2}$, where $g$ is the amount of gold spent (since 2 gold = 5 shops).