Codeforces Round 730 (Div. 2) |
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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,
For example,
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.
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.
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 $$$\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$$$.
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
1.532000000000 1.860000000000 5.005050776521 4.260163673896
For the first test case, the possible drawing sequences are:
For the second test case, the possible drawing sequences are:
So, the expected number of races is equal to $$$1\cdot 0.4 + 2\cdot 0.24 + 3\cdot 0.16 + 2\cdot 0.1 + 3\cdot 0.1 = 1.86$$$.
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