| <p>You may be familiar with the works of Alfred, Lord Tennyson, the famous | |
| English poet. In this problem we will concern ourselves with Tennison, | |
| the less famous English tennis player. As you know, tennis is not so much | |
| a game of skill as a game of luck and weather patterns. The goal of tennis is | |
| to win <strong>K</strong> sets before the other player. However, the chance of winning a set is | |
| largely dependent on whether or not there is weather. | |
| </p> | |
| <p>Tennison plays best when it's sunny, but sometimes, of course, it rains. | |
| Tennison wins a set with probability <strong>p<sub>s</sub></strong> when it's sunny, and with probability | |
| <strong>p<sub>r</sub></strong> when it's raining. The chance that there will be sun for the first set is | |
| <strong>p<sub>i</sub></strong>. Luckily for Tennison, whenever he wins a set, the probability that there | |
| will be sun increases by <strong>p<sub>u</sub></strong> with probability <strong>p<sub>w</sub></strong>. Unfortunately, when Tennison | |
| loses a set, the probability of sun decreases by <strong>p<sub>d</sub></strong> with probability <strong>p<sub>l</sub></strong>. | |
| What is the chance that Tennison will be successful in his match?</p> | |
| <p>Rain and sun are the only weather conditions, so P(rain) = 1 - P(sun) at all | |
| times. Also, probabilities always stay in the range [0, 1]. If P(sun) would | |
| ever be less than 0, it is instead 0. If it would ever be greater than 1, it | |
| is instead 1.</p> | |
| <h2>Input</h2> | |
| <p> | |
| Input begins with an integer <strong>T</strong>, the number of tennis matches | |
| that Tennison plays. For each match, there is a line containing an integer | |
| <strong>K</strong>, followed by the | |
| probabilities <strong>p<sub>s</sub>, p<sub>r</sub>, p<sub>i</sub>, p<sub>u</sub>, p<sub>w</sub>, p<sub>d</sub>, p<sub>l</sub></strong> in that order. All of these | |
| values are given with exactly three places after the decimal point. | |
| <h2>Output</h2> | |
| <p> | |
| For each match, output "Case #i: " followed by the probability that Tennison wins the match, rounded to 6 decimal places | |
| (quotes for clarity only). It is guaranteed that the output is unaffected by deviations as large as 10<sup>-8</sup>. | |
| </p> | |
| <h2>Constraints</h2> | |
| <ul> | |
| <li>1 ≤ <strong>T</strong> ≤ 100</li> | |
| <li>1 ≤ <strong>K</strong> ≤ 100</li> | |
| <li>0 ≤ <strong>p<sub>s</sub>, p<sub>r</sub>, p<sub>i</sub>, p<sub>u</sub>, p<sub>w</sub>, p<sub>d</sub>, p<sub>l</sub></strong> ≤ 1</li> | |
| <li><strong>p<sub>s</sub></strong> > <strong>p<sub>r</sub></strong></li> | |
| </ul> | |