Calculating the Elo rating if x beats y z % of the time

Question

From 2015 - 2019 dancers competing at the Lindy Hope Championships were split into 4 levels of expertise. The level four dancers beat the level three dancers on average 83% of the time. If we were to automatically assume that level four dancers have a higher elo rating than level three dancers how much higher would that rating be. The equation for calculating elo is: $$ e = \frac{1} {1 + 10^{(b-a)/400}}, $$ where $e$, I think, is the expected amount of time of victory. $b$ is the rating of the level 4 dancer. $a$ is the rating of the level 4 dancer.

We know that $e$ must be $e= 0.83$ and $a=0$. But I cannot figure out how to solve for $b$. Any insight would be appreciated.

Answer 1

The solution is as follows

\begin{align}e &= \frac{1} {1 + 10^{(b-a)/400}}\\ e\left(1+10^{(b-a)/400}\right)&=1\tag{1}\\ 1+10^{(b-a)/400}&=\frac 1e\tag{2}\\ 10^{(b-a)/400}&=\frac1e -1\tag{3}\\ \frac{b-a}{400}&=\log_{10}\left(\frac 1e -1\right)\tag{4}\\ b-a&=400\log_{10}\left(\frac 1e -1\right)\tag{5}\\ b&=400\log_{10}\left(\frac 1e -1\right)+a\tag{6}\end{align}

where the steps are

1. Multiply both sides by $e$
2. Divide both sides by $1+10^{(b-a)/400}$
3. Subtract $1$ from both sides
4. Take the logarithm of both sides. If we have $x=10^y$ then we can say that $\log_{10}(x)=y$. Here is a good introduction to logarithms
5. Multiply both sides by $400$
6. Add $a$ to both sides

You can then use Wolfram|Alpha to solve the equation by filling in your values for $a$ and $e$ to find that $$b=-275.452$$