2017/quals/fightingthezombie.html

<p>
<strong>"Okay, Wizard, cast your spell!"</strong>
</p>

<p>
But which of your many spells to cast? In the ever-popular role-playing game 
<em>Dungeons & Dragons</em>, or <em>D&D</em>, you determine a spell's damage 
by rolling polyhedral
dice with 4, 6, 8, 10, 12, or 20 sides. Since there's a lot of dice-rolling
involved, players use shorthand to denote which dice should be rolled.
<strong>X</strong>d<strong>Y</strong> means 
"roll a <strong>Y</strong>-sided die <strong>X</strong> times, and sum the rolls''. 
Sometimes, you must add or subtract a value <strong>Z</strong> after
you finish rolling, in which case the notation is 
<strong>X</strong>d<strong>Y</strong>+<strong>Z</strong> or 
<strong>X</strong>d<strong>Y</strong>-<strong>Z</strong> respectively.
</p>

<p>
For example, if you roll 2d4+1, you'll end up with a result between 3 and 9
inclusive. If you roll 1d6-3, your result will be between -2 and 3 inclusive.
</p>

<p>
In <em>D&D</em>, wizards are powerful but flimsy spellcasters. As a wizard
fighting a zombie, your best strategy is to maximize the chance that you can
kill the zombie with a single spell before it has a chance to retaliate. What
spell should you cast?
</p>



<h3>Input</h3>

<p>
Input begins with an integer <strong>T</strong>, the number of zombies
you'll fight. For each zombie, there are two lines. The first contains two 
integers, <strong>H</strong> and <strong>S</strong>, the minimum amount of
damage it takes to defeat the zombie, and the number of spells you have prepared, 
respectively. The second line contains <strong>S</strong> spell descriptions separated by
single spaces. A spell description is simply the amount of damage a spell does
in the notation described above. 
</p>



<h3>Output</h3>

<p>
For each zombie, print a line containing the probability of defeating the zombie if you select your spell optimally.
</p>

<p>
Absolute and relative errors of up to 1e-6 will be ignored.
</p>

<h3>Constraints</h3>

<p>
1 &le; <strong>T</strong> &le; 1,000 <br />
1 &le; <strong>H</strong> &le; 10,000 <br />
2 &le; <strong>S</strong> &le; 10 <br />
</p>

<p>
Additionally, the following constraints will hold for each spell:
</p>

<p>
1 &le; <strong>X</strong> &le; 20 <br />
<strong>Y</strong> &isin; {4, 6, 8, 10, 12, 20} <br />
1 &le; <strong>Z</strong> &le; 10,000, if <strong>Z</strong> is specified. <br />
<strong>X</strong>, <strong>Y</strong>, and <strong>Z</strong> 
will be integers with no leading zeros. <br />
</p>

<h3>Explanation of Sample</h3>

<p>
In the first case, you can guarantee a kill with the first spell, which must always do at least 2 damage.
</p>

<p>
In the third case, your first spell is the best. If you roll a 4, you'll do the requisite 8 damage. The second spell requires
rolling a 4 on two dice rather than just one, and the third spell requires rolling a 4 on all three dice.
</p>