Combat Odds

On game forums there is always someone who blames the game for his own losses. A lack of realism or a cheating AI are the usual charges. Most of the time the real problem is skill.

The latest complaint comes from a player who says the computer cheats in the game he just bought. A battle is reported as a 70% chance to win and he loses it anyway. Another player takes his side and adds that he has lost five such battles in a row. The probability of that is $(1 - 0.7)^5 = 0.00243$, so the game cannot be honest, he says. That number is right only if those five battles are the only ones that ever happen. A single game contains far more battles than five, so the chance that such a streak shows up somewhere is much larger. You decide to compute it yourself.

A battle ends in a win or a loss, and the probability of a win is $p$. Given $N$ battles, find the probability that a losing streak of length at least $L$ appears at least once. Battle outcomes are independent of each other and the random number generator is fair.

The first line contains the number of test cases $T$. Each of the next $T$ lines contains $N$, $L$ and $p$ for one test case, separated by whitespace.

• $0 < T \le 200$
• $0 < N \le 2000$
• $0 < L \le N$
• $0 \le p \le 1.0$

For each test case, print on its own line the probability that a losing streak of length at least $L$ appears. Round the value to exactly nine digits after the decimal point and pad with zeros when digits are missing.