Practice

We want to model how a team's chance of winning a programming contest depends on how much it has practiced.

Let $p$ be the probability that a given team wins a given contest, and let $n$ be the number of practice problems the team solved before the contest. We assume the two are related by the logistic model

$\log\frac{p}{1-p} = a + b\,n$

for some constants $a$ and $b$. Your task is to find the $a$ and $b$ for which this model best fits a set of observed results.

Each observation is a pair $(n, w)$: $n$ is the number of practice problems a team solved before a contest, and $w$ is $1$ if the team won that contest and $0$ otherwise.

Given $a$, $b$, and $n$, the model yields $p$, the estimated probability that $w = 1$. The likelihood of a single observation is $p$ when $w = 1$ and $1 - p$ when $w = 0$; the likelihood of a set of observations is the product of the likelihoods of the individual observations.

Compute the maximum-likelihood estimate of $a$ and $b$ — the values that maximize the likelihood of the given set of observations.

The input contains several test cases, followed by a line containing a single $0$.

Each test case begins with an integer $k$ ($1 < k \le 100$), the number of observations that follow. Each of the next $k$ lines contains two integers $n$ and $w$ ($0 \le n \le 100$, $0 \le w \le 1$). Within each test case there are at least two distinct values of $n$ and at least two distinct values of $w$.

For each test case, print a single line containing $a$ and $b$, each rounded to four digits to the right of the decimal point.