If Player \(A\) has a rating of \(R_{A}\) and Player \(B\) a rating of \(R_{B}\), the exact formula (using the logistic curve) for the expected score of Player \(A\) is \[E_A = \frac{1}{1+10^{\frac{R_B-R_A}{400}}} \]
Similarly the expected score for Player \(B\) is
\[E_B = \frac{1}{1+10^{\frac{R_A-R_B}{400}}} \]
This could also be expressed by
\[E_A = \frac{Q_A}{Q_A + Q_B} \]
\[E_B = \frac{Q_B}{Q_A + Q_B} \]
where \(Q_A = 10^{\tfrac{R_A}{400}}\) and \(Q_B = 10^{\tfrac{R_B}{400}}\)
Supposing Player \(A\) was expected to score \(E_{A}\) points but actually scored \(S_{A}\) points. The formula for updating their rating is \[R_{A}^{\prime}=R_{A}+K(S_{A}-E_{A}) \]