K. Uniform Chemistry
dp; math
In a parallel universe there are \(n\) chemical elements, numbered from \(1\) to \(n\). The element number \(n\) has not been discovered so far, and its discovery would be a pinnacle of research and would bring the person who does it eternal fame and the so-called SWERC prize.There are \(m\) independent researchers, numbered from \(1\) to \(m\), that are trying to discover it. Currently, the \(i\)-th researcher has a sample of the element \(s_i\). Every year, each researcher independently does one fusion experiment. In a fusion experiment, if the researcher currently has a sample of element \(a\), they produce a sample of an element \(b\) that is chosen uniformly at random between \(a+1\) and \(n\), and they lose the sample of element \(a\). The elements discovered by different researchers or in different years are completely independent.The first researcher to discover element \(n\) will get the SWERC prize. If several researchers discover the element in the same year, they all get the prize. For each \(i = 1, \, 2, \, \dots, \, m\), you need to compute the probability that the \(i\)-th researcher wins the prize.
The first line contains two integers \(n\) and \(m\) (\(2 \le n \le 10^{18}\), \(1 \le m \le 100\)) — the number of elements and the number of researchers.The second line contains \(m\) integers \(s_1, \, s_2, \, \dots, \, s_m\) (\(1 \le s_i < n\)) — the elements that the researchers currently have.
Print \(m\) floating-point numbers. The \(i\)-th number should be the probability that the \(i\)-th researcher wins the SWERC prize. Your answer is accepted if each number differs from the correct number by at most \(10^{-8}\).
In the first sample, all researchers will discover element \(2\) in the first year and win the SWERC prize.In the second sample, the last researcher will definitely discover element \(3\) in the first year and win the SWERC prize. The first two researchers have a \(50\%\) chance of discovering element \(2\) and a \(50\%\) chance of discovering element \(3\), and only element \(3\) will bring them the prize.In the third sample, each researcher has an independent \(50\%\) chance of discovering element \(3\) in the first year, in which case they definitely win the SWERC prize. Additionally, if they all discover element \(2\) in the first year, which is a \(12.5\%\) chance, then they will all discover element \(3\) in the second year and all win the prize.