308587: CF1543C. Need for Pink Slips
Memory Limit：256 MB
Time Limit：1 S
Judge Style：Text Compare
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Description
Need for Pink Slips
题意翻译
在完成一项游戏任务后，你得到了一次抽奖的机会。 初始时，抽奖只有三种**有效**奖励：Cash Slip,Impound Strike Release Marker 和 Pink Slip，初始时你抽到它们的概率为 $c,m,p$，同时抽奖还有一个不稳定性 $v$。你将会一直抽奖直到抽中一张 Pink Slip，假设在一次抽奖中你抽到了某个被抽中概率为 $a$ 的奖励，接下来： - 如果你抽中的是 Pink Slip，停止抽奖。 - 否则，如果 $a\leq v$，抽中这种奖励的概率变为 $0$，且这种奖励将失效，并永远不会出现在接下来的抽奖中，这种奖励被抽到的概率 $a$ 将平均分配给剩下的**还有效的**奖励的概率中。 - 否则，$a$ 将减少 $v$，减少的概率同样将平均分配给剩下的**除了这种奖励外还有效的**奖励的概率中。 求出期望的抽奖次数。$T$ 组数据。 $1\leq T\leq10;0<c,m,p<1;c+m+p=1;0.1\leq v\leq0.9;$ $c,m,p,v$ 最多只有 $4$ 位小数。 **你的答案与标准答案的相对或绝对误差不能超过 $10^{-6}$。** --- 如果你无法理解抽奖的过程，下面是关于抽奖的一些例子： - 如果 $(c,m,p)=(0.2,0.1,0.7),v=0.1$，且你抽到了 Cash Slip，抽奖后 $(c,m,p)=(0.1,0.15,0.75)$。 - 如果 $(c,m,p)=(0.1,0.2,0.7),v=0.2$，且你抽到了 Cash Slip，抽奖后 $(c,m,p)=(0,0.25,0.75)$，Cash Slip 将失效。 - 如果 $(c,m,p)=(0.2,0,0.8),v=0.1$，且你抽到了 Cash Slip，抽奖后 $(c,m,p)=(0.1,0,0.9)$。 - 如果 $(c,m,p)=(0.1,0,0.9),v=0.2$，且你抽到了 Cash Slip，抽奖后 $(c,m,p)=(0,0,1)$，Cash Slip 将失效。题目描述
After defeating a Blacklist Rival, you get a chance to draw $ 1 $ reward slip out of $ x $ hidden valid slips. Initially, $ x=3 $ and these hidden valid slips are Cash Slip, Impound Strike Release Marker and Pink Slip of Rival's Car. Initially, the probability of drawing these in a random guess are $ c $ , $ m $ , and $ p $ , respectively. There is also a volatility factor $ v $ . You can play any number of Rival Races as long as you don't draw a Pink Slip. Assume that you win each race and get a chance to draw a reward slip. In each draw, you draw one of the $ x $ valid items with their respective probabilities. Suppose you draw a particular item and its probability of drawing before the draw was $ a $ . Then, - If the item was a Pink Slip, the quest is over, and you will not play any more races. - Otherwise, 1. If $ a\leq v $ , the probability of the item drawn becomes $ 0 $ and the item is no longer a valid item for all the further draws, reducing $ x $ by $ 1 $ . Moreover, the reduced probability $ a $ is distributed equally among the other remaining valid items. 2. If $ a > v $ , the probability of the item drawn reduces by $ v $ and the reduced probability is distributed equally among the other valid items. For example, - If $ (c,m,p)=(0.2,0.1,0.7) $ and $ v=0.1 $ , after drawing Cash, the new probabilities will be $ (0.1,0.15,0.75) $ . - If $ (c,m,p)=(0.1,0.2,0.7) $ and $ v=0.2 $ , after drawing Cash, the new probabilities will be $ (Invalid,0.25,0.75) $ . - If $ (c,m,p)=(0.2,Invalid,0.8) $ and $ v=0.1 $ , after drawing Cash, the new probabilities will be $ (0.1,Invalid,0.9) $ . - If $ (c,m,p)=(0.1,Invalid,0.9) $ and $ v=0.2 $ , after drawing Cash, the new probabilities will be $ (Invalid,Invalid,1.0) $ . You need the cars of Rivals. So, you need to find the expected number of races that you must play in order to draw a pink slip.输入输出格式
输入格式
The first line of input contains a single integer $ t $ ( $ 1\leq t\leq 10 $ ) — the number of test cases. The first and the only line of each test case contains four real numbers $ c $ , $ m $ , $ p $ and $ v $ ( $ 0 < c,m,p < 1 $ , $ c+m+p=1 $ , $ 0.1\leq v\leq 0.9 $ ). Additionally, it is guaranteed that each of $ c $ , $ m $ , $ p $ and $ v $ have at most $ 4 $ decimal places.
输出格式
For each test case, output a single line containing a single real number — the expected number of races that you must play in order to draw a Pink Slip. Your answer is considered correct if its absolute or relative error does not exceed $ 10^{-6} $ . Formally, let your answer be $ a $ , and the jury's answer be $ b $ . Your answer is accepted if and only if $ \frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6} $ .
输入输出样例
输入样例 #1
4
0.2 0.2 0.6 0.2
0.4 0.2 0.4 0.8
0.4998 0.4998 0.0004 0.1666
0.3125 0.6561 0.0314 0.2048输出样例 #1
1.532000000000
1.860000000000
5.005050776521
4.260163673896