TopCoder Statistics - Problem Statement

Problem Statement

You are playing a game in a casino. The game is played with a sequence of positive real numbers. You are given a int[] S. For each valid i, element i of the initial sequence can be computed as S[i] / 1000.0.

During the game, some elements of the sequence will become erased. Once an element has been erased, it remains erased for the rest of the game. The game ends when all elements have been erased.

Your initial score is zero.

The game is played in turns. Each turn has three phases:

An element of the sequence that hasn't been erased yet is chosen uniformly at random.
That element is added to your score and then erased.
A random effect is applied to each element that hasn't been erased yet.

For each element, the random effect that will be applied is randomly chosen as follows:

With probability 33% the element is erased (without being added to your score).
With probability 33% the element is divided by two. (If its current value is x, its new value will be x/2.)
With probability 33% the element is replaced by its square root. (If its current value is x, its new value will be sqrt(x).)
With probability 1% the element is restored to its initial value. (Regardless of its current value, its new value will be the value it had before the first turn of the game.)

The random effects for different elements are mutually independent.

Return the expected value of your score at the end of the game.

Definition

Class:: CasinoGame
Method:: expectedValue
Parameters:: int[]
Returns:: double
Method signature:: double expectedValue(int[] S)
(be sure your method is public)

Notes

The returned value must have an absolute or relative error less than 1e-9.

Constraints

S will contain between 1 and 1,000 elements, inclusive.
Each element of S will be between 1 and 1,000,000, inclusive.

Examples

{100000, 100000}

Returns: 120.80000000000001

We start the game with the sequence {100, 100}. Regardless of which element is chosen in the first step, our score will increase by 100 and that element will be erased. Let's consider what will happen to the other element: With probability 33% the other element is also erased, the game ends, and our score is 100. With probability 33% the element becomes 50. In the second turn we add it to our score and we end the game with the score 150. With probability 33% the element becomes 10. In the second turn we add it to our score and we end the game with the score 110. With probability 1% the element is restored to its initial value, which means that it will remain 100. In this case our final score will be 200. Thus, our expected final score is 0.33*100 + 0.33*150 + 0.33*110 + 0.01*200.
{68391}

Returns: 68.391
{9000, 25000, 100000}

Returns: 59.47600366439891

The initial sequence is {9, 25, 100}. One possible gameplay looks as follows: Element 1 is selected, added to your score, and erased. Element 0 is halved and element 2 is replaced by its square root. The new sequence is {4.5, erased, 10}. Element 0 is selected, added to your score, and erased. Element 1 is already erased, so nothing happens there. Element 2 is restored to its original value. The new sequence is {erased, erased, 100}. Element 2 is selected, added to your score, and erased. The game ends. In this game your final score was 25 + 4.5 + 100 = 129.5.
{932452, 32431, 234511, 1, 999, 12341, 432134, 987654, 1000000}

Returns: 569.7984079363877
{655482,657692,166545,510808,406135,487463,572526,89578,844121,881092,843697,733297,998007,518962,879347,652488,193968,569275,486335,194222,498550,98979,729632,518857,959392,22656,728919,158737,793281,75238,760687,449635,732925,558907,592118,770734,47242,165516,491986,891358,678283,336555,257207,308841,487191,137426,961324,681154,338375,80210, 875371,468599,809986,237554,987452,770250,891008,348922,928983,685161,424155,690542,766471,788754,881123,359461,560360,928360,156651,684020,452270,466608,21447,341151,775444,140312,110251,369319,821461,448621,81203,329383,548895,891185,566933,168898,662307,458813,517815,223841,775648,573645,546058,174670,363271,428053,165805,555306,357286,322452, 240198,441230,420734,892443,414055,827853,33627,524301,828846,855084,604596,910045,816141,154363,802102,15625,323256,96960,106112,472746,320797,881756,47263,866850,688100,410529,295775,853901,597509,284735,808027,837703,357639,229633,362697,771689,58358,396319,927664,518879,883077,533133,429796,700091,687491,863572,715711,642422,592206,821819, 116040,912998,336126,794100,412399,25098,836303,339849,510673,434685,256258,319572,904062,613893,180880,267631,18133,239233,663945,945793,389786,179574,111477,819577,879660,430642,315700,227922,73936,907902,681415,820773,453451,649215,615745,865846,674309,452921,837369,816656,519280,94499,137101,424214,340066,317976,691840,988997,188883,386109}

Returns: 817.8156226417173
{1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000, 1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000,1000000}

Returns: 1650.8120310299312

Here S has 1,000 elements, all equal to 1,000,000.
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

Returns: 3.2126045329902637
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Returns: 3.3218797114341125
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Returns: 1248.7173792397211
{57,59,21,51,57,71,7,1,58,32,97,24,2,77,25,67,61,35,31,66,65,48,39,22,2,31,57,89,83,94,2,92,52,75,42,61,45,48,61,54,79,57,77,32,85,53,98,97,88,28,62,4,28,52,26,29,82,82,17,65,75,71,56,78,97,49,38,93,97,98,46,27,6,74,59,42,27,56,39,14,36,52,69,63,4,94,43,37,75,60,1,49,82,8,27,78,9,16,70,5,14,67,31,71,40,41,13,18,97,3,83,84,54,52,98,9,45,40,46,20,51,98,20,84,58,46,61,66,62,82,70,27,48,52,49,40,93,13,57,41,15,40,76,21,91,73,81,87,64,26,6,67,76,78,50,33,75,63,50,36,44,71,14,44,74,15,83,18,27,91,58,94,82,85,66,72,57,46,11,73,24,68,39,99,45,88,83,20,50,32,7,46,54,73,89,27,87,23,45,65,65,54,58,47,39,23,70,47,21,80,19,44,48,9,42,92,97,24,63,98,7,70,43,12,42,83,38,28,57,34,44,22,88,2,68,78,76,37,24,96,69,95,91,16,3,32,59,51,7,22,49,65,43,91,76,84,26,66,63,82,99,6,3,38,59,22,67,87,11,91,34,79,37,25,46,91,8,4,42,15,77,90,31,19,32,7,2,9,24,16,91,74,22,45,12,32,19,78,70,29,20,4,7,8,80,52,99,87,7,92,1,84,81,32,2,64,90,56,73,65,71,15,38,44,11,1,76,29,31,45,57,2,100,63,10,79,66,60,66,73,51,18,56,83,1,9,98,90,64,70,6,35,36,44,30,47,44,5,75,74,2,32,28,53,46,37,32,12,48,49,36,50,66,91,32,67,51,81,8,15,51,14,1,38,9,30,84,52,87,59,78,40,42,5,92,39,93,23,2,92,71,37,41,89,79,24,55,30,4,62,44,6,27,44,44,35,25,79,39,11,89,16,50,30,72,42,21,16,16,22,59,39,11,99,27,89,74,81,18,29,94,13,35,73,8,30,7,33,8,97,95,97,64,97,78,87,38,98,2,5,72,60,43,82,10,21,70,83,53,40,12,99,4,98,71,64,27,29,48,86,26,42,34,41,38,12,28,27,61,29,84,84,41,78,65,2,99,87,37,51,78,48,49,81,97,71,44,75,100,91,60,77,85,46,17,74,57,44,53,69,25,36,53,17,13,69,18,63,55,54,66,32,53,66,65,49,37,60,75,88,3,87,64,87,32,32,12,40,28,64,60,52,51,12,68,64,81,37,78,87,91,43,71,43,9,35,44,97,94,18,84,96,56,99,34,87,30,46,78,57,9,38,60,60,49,79,75,81,16,4,20,6,47,90,100,7,24,95,3,17,65,86,65,20,36,98,7,65,43,84,74,4,21,33,63,22,12,89,54,79,92,73,36,90,62,35,96,85,30,50,54,46,87,70,65,22,67,23,87,62,59,12,65,31,44,27,52,7,67,58,85,58,30,20,48,44,7,95,80,88,45,85,33,31,54,49,53,21,72,91,82,30,54,98,60,49,76,64,56,42,21,92,51,2,64,98,97,70,45,77,9,89,13,41,71,67,89,23,39,12,65,72,41,18,69,53,67,44,68,74,85,40,17,87,41,80,85,90,1,81,18,9,69,30,1,39,96,42,14,34,53,30,5,46,100,25,50,18,68,17,91,4,56,7,91,48,39,27,37,39,7,54,100,27,36,52,65,31,93,30,17,98,60,73,95,59,98,44,76,17,12,18,21,67,76,63,66,14,41,55,5,47,60,56,25,95,7,89,26,52,71,94,1,30,66,47,40,63,90,67,32,53,84,4,71,11,66,36,77,6,90,81,4,50,36,28,96,94,68,21,45,38,66,97,19,32,43,58,46,84,76,77,36,11,80,6,22,97,94,98,54,35,30,57,36,17,36,32,10,56,4,7,93,70,3,64,1,46,73,98,29,49,75,17,59,6,74,32,3,67,81,8,2,62,65,37,78,52,68,40,7,24,98,52,93,52,15,45,49,87,42,30,87,68,46,98,74,19,81,28,86,62,35,39,75,51,75,53,3,95,44,61,18,93,64,62,44,30,6,93,69,47,22,55,15,67,4,40,37,85,67,22,98,53,12,72,4,39,76,58,85,19,70,54,11,34,15,7,63,20,99,31,18,72,38,84,38,41,75,26,25,41,100,74,46,11,46,49,49,21,58,33,92,27,38,54,60,52,60,23,23,58,5,93,81,42,28,70,83,3,96,59,95,95,33,40,5,30,88,6}

Returns: 4.231740201136852
{37,48,29,48,47,12,41,40,45,23,11,30,11,25,26,46,24,31,44,41,22,29,37,44,18,11,45,12,25,28,13,13,25,35,13,23,37,44,15,33,18,16,14,21,33,30,19,50,13,12,40,27,31,28,20,41,31,15,43,48,35,49,13,50,33,16,25,21,50,32,46,19,40,10,31,23,32,40,24,37,44,13,16,26,34,26,16,14,33,10,14,17,10,17,19,35,23,36,46,24,19,44,33,49,46,15,23,29,45,37,16,38,43,24,15,26,42,21,32,27,24,36,36,24,43,45,50,17,32,47,31,41,40,16,42,37,21,14,16,16,44,24,46,36,38,11,13,29,24,35,46,40,23,32,14,17,28,15,25,12,11,48,45,44,13,36,32,27,43,38,35,36,13,30,21,41,33,26,22,50,10,19,39,25,43,45,33,21,11,50,25,13,47,21,49,12,48,30,29,42,20,13,27,23,35,40,16,20,15,28,19,17,37,50,33,30,47,17,43,48,16,19,12,14,30,12,16,29,33,37,20,45,40,37,17,27,28,23,37,33,43,48,41,32,47,23,13,45,32,46,43,40,16,47,44,36,50,12,17,34,39,27,28,31,16,37,48,34,12,36,17,45,33,50,26,31,24,29,26,46,26,18,35,34,16,28,20,15,30,29,39,21,46,18,42,13,46,41,38,48,26,47,44,49,46,22,31,19,41,47,16,19,16,41,43,23,20,14,28,43,33,19,13,31,27,47,34,24,39,23,23,16,21,19,16,16,31,37,27,23,34,33,32,40,25,27,12,36,31,33,30,16,42,35,37,20,31,20,37,21,36,11,27,47,20,33,13,41,19,32,16,43,17,38,35,32,16,39,19,38,21,41,44,14,25,30,27,46,42,15,18,27,16,35,24,29,17,29,21,29,12,27,21,19,17,48,44,23,37,14,10,48,46,46,14,20,27,31,18,21,38,26,38,44,13,13,22,20,34,34,41,37,12,12,48,19,11,43,33,38,48,35,38,45,31,42,14,50,24,24,20,13,43,10,47,46,15,21,17,40,47,49,28,49,12,25,18,13,18,43,44,17,27,31,11,50,24,17,50,40,34,19,43,26,19,39,23,27,11,32,16,48,32,34,49,34,10,18,40,20,10,33,27,27,15,30,29,29,38,28,18,23,37,10,39,49,41,13,27,45,35,33,44,17,18,42,43,20,11,32,31,11,16,50,31,21,32,50,40,19,27,50,32,15,11,22,15,45,25,32,39,11,16,32,18,24,26,11,37,27,33,17,31,42,18,11,12,40,12,43,10,31,42,34,38,46,48,44,40,22,27,28,24,34,11,34,50,27,35,38,47,20,47,27,11,15,28,13,47,32,48,48,12,42,33,40,37,31,35,26,45,12,46,18,38,47,45,39,26,29,27,22,41,23,39,44,28,18,50,25,40,47,24,42,40,47,33,28,29,18,44,24,22,41,34,50,40,28,38,15,50,14,29,42,28,17,36,46,27,35,22,16,33,38,50,25,37,34,43,15,42,39,31,13,29,15,12,18,35,43,25,34,47,44,26,24,12,13,22,29,40,36,37,24,24,36,39,12,20,32,17,13,20,39,17,41,46,21,11,30,13,28,16,12,22,34,26,26,37,40,46,28,28,32,42,44,20,31,46,32,14,14,35,26,45,44,16,40,17,19,22,20,38,28,24,11,11,41,29,40,32,24,17,50,49,10,43,20,33,40,42,37,45,28,12,39,22,21,31,29,32,43,41,19,22,15,22,25,48,42,16,31,17,25,33,17,25,25,27,49,15,18,35,11,39,40,40,10,12,22,31,34,16,23,44,28,30,17,45,29,49,10,12,18,25,35,25,41,50,43,41,16,12,27,19,41,16,50,43,18,23,23,43,30,39,38,50,20,45,45,40,46,45,42,15,22,28,30,14,29,24,45,36,27,21,45,17,30,46,12,40,21,25,32,41,15,21,42,26,18,36,17,15,33,10,20,47,28,42,10,47,16,47,34,33,19,31,42,39,26,44,29,39,21,12,31,28,26,23,44,34,10,12,41,35,12,10,31,30,42,33,29,48,29,12,32,39,33,26,29,11,19,50,40,32,14,21,10,30,36,46,15,36,49,46,21,12,48,44,35,39,27,13,39,48,17,22,38,41,38,19,42,50,18,33,33,22,46,35,44,31,31,10,17,31,46,30,35,43,25,19,34,44,24,24,42,32,38,31,22,28,40,15,27,10,38,11,24,34,37,18,14,19,20,23,42,16,45,26,10}

Returns: 4.0510704362234575
{488805,161560,965084,738511,33771,263101,500382,509767,724029,277312,491227,852529,579453,483792,572440,56630,859076,564277,223263,256887,336063,337739,664723,523247,621080,898863,513807,600716,573279,951460,194109,578435,629371,675545,316945,663142,454997,333679,689260,179025,610990,180486}

Returns: 757.3452945945535
{190443,664277,603993,763424,523352,684621,503038,296591,537035,357128,961313,60282,978207,860176,90440,95275,949806,41899,289383,528240,671270,481279,361536,850763,936276,695214,540022,115300,306204,236859,663205,12998,901135,783550,292773,424486,468170,312162,237428,521557,669289,198741,581838,647496,575268,188629,259122,525073,230528,548504,53312,418149,29783,414847,785263,482410,110061,841636,114061,932616,78494,777266,461965,495980,560815,754737,920465,545336,66898,674245,583244,252538,872985,681433,900033,964604,870062,675506,489676,100589,224010,59339,35089,770144,474185,336703,252553,100597,694690,366613,33212,289535,660230,11528,785514,737396,282616,222330,799084,865865,896574,382327,118403,285910,63760,534787,250513,933821,210293,256540,550761,950654,315878,102201,720797,790063,955255,489701,407011,649944,372665,956575,455830,32895,968102,241343,286642,250718,980024,85725,116582,392950,468052,751336,678859,48163,286123,445724,981983,496415,702263,49095,963420,534493,151295,684216,324555,106549,690268,731565,272844,62932,688139,728673,612178,172593,486367,898820,423310,982742,984544,56243,375691,968947,807579,570902,533461,610053,16625,515443,106467,235239,80889,586238,769731,232183,270453,610637,855083,477072,858554,127926,540003,63044,372950,668533,235636,375668,567352,175297,358410,68247,231540,734100,553546,555470,305001,87006,165522,837977,118801,788340,73216,199689,374577,842946,948224,161381,969935,803306,638452,828488,447584,694806,891531,820533,363338,127167,196201,447041,302463,554610,31640,50354,805061,585185,605823,110062,188542,771344,948038,307342,76035,537605,23383,450611,896903,971606,611991,866837,774911,766794,695324,222494,461600,103206,559379,341289,746724,755579,788330,49187,826540,819969,615892,631600,921505,738067,258013,626398,25762,722403,450092,101797,776359,473474,552407,673261,445079,680750,56449,736341,447543,268124,475187,425494,371330,34565,766783,118053,306495,71464,683591,133034,407784,815835,280985,845640,553901,55350,472037,579662,294104,922128,681458,70462,395601,750217,743723,357031,947318,316523,609724,911212,584647,84910,336706,955976,119474,619840,590380,942320,691303,790323,75353,615438,606157,872689,461077,160057,444390,933113,739718,738493,855241,937528,808955,767193,204096,69029,640576,151413,385551,250299,578976,970197,851560,915681,442524,971033,51872,549256,913352,259526,339578,505056,874963,945734,894096,336039,622142,338486,269152,878211,76978,640744,332090,885932,924288,536185,954960,81215,203949,856863,331513,782925,343411,183072,214957,785935,670456,266829,851542,100159,526354,191119,121566,917669,653204,15662,253707,791697,870499,39210,669907,947476,679953,1997,349760,120593,54533,821071,201807,258482,677933,49672,41406,537696,749095,772714,323630,935903,39542,175171,36061,82248,882641,673979,999916,52196,205992,769974,843892,76490,809184,30150,540317,5488,548498,406428,126080,603031,227499,844239,377864,421783,893910,935621,959478,643004,708334,799459,578906,264228,490981,131319,862827,889973,321649,379094,942168,527640,149067,302411,604129,958250,332561,660797,480090,397410,67225,122521,516792,294723,966759,894655,232857,377020,346627,708687,536376,571313,508145,631633,835540,515478,279303,698366,405450,600951,77459,863970,128590,742877,166380,249070,701127,15292,909867,697568,412702,977091,820088,445845,788165,303199,856852,537373,196570,203478,246059,732945,774790,270556,364578,126681,302385,643880,341398,707834,761183,418856,88155,889772,678085,254535,655194,895563,786178,565060,593130,715231,58502,929569,161076,363018,232767,17927,416742,429337,737756,179153,678633,28898,966060,559562,155578,268444,719794,496976,492629,480976,432183,580784,370747,110267,835318,542292,5829,621495,107351,115310,336726,682204,44879,14153,561573,793997,548431,978315,223333,286186,673819,418318}

Returns: 809.2423215454543
{639878,977879,470661,908321,697672,483988,917301,178647,432522,498084,65746,542789,333401,608037,64969,471248,231740,180279,324325,430295,741509,854829,991868,535505,403259,486534,275190,689444,160352,693507,520879,316581,671385,507891,741253,369057,991878,658553,64055,424399,156637,129800,483539,6389,254189,64860,993988,485928,761490,834664,432574,502998,689492,424441,38502,92750,427326,313691,298546,104029,523549,819424,420609,194934,327314,678213,80342,835543,336766,144396,776293,493402,790548,776184,16142,44736,841043,10130,47015,602532,844793,479588,621881}

Returns: 720.0973385896381
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Returns: 771.5805018811948
{864352,960377,640934,683531,197479,870153,230333,398756,229464,528471,957750,739013,61658,205072,461690,579155,80788,433683,817327,70888,895492,29304,690781,366084,72950,665433,415491,844762,325437,344242,145888,706141,304618,303174,906023,18448,173326,136356,417204,402789,664826,891305,658154,242835,612728,119843,821990,693515,69877,639316,764402,481720,184971,971535,364155,257920,636967,295998,619033,478756,640239,281273,184896,461208,584446,90918,996008,757771,743625,929563,676912,408450,337219,335065,651285,949946,971259,473274,643460,41135,628941,924213,39206,813911,895747,403361,588182,49066,699358,207214,44173,855948,488486,229068,833507,72931,836337,345866,347054,579962,275428,23965,504763,128998,875381,156047,78943,846639,145672,238754,404125,774612,162967,443330,104874,575065,363042,693055,140482,62399,416621,184654,434698,905106,930073,268205,978037}

Returns: 748.6656258690447
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Returns: 794.1770410286251
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Returns: 854.6520388481117
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Returns: 834.835759535611
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Returns: 791.5801647107089
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Returns: 776.6734482756827
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999501, 999500, 999499, 999498, 999497, 999496, 999495, 999494, 999493, 999492, 999491, 999490, 999489, 999488, 999487, 999486, 999485, 999484, 999483, 999482, 999481, 999480, 999479, 999478, 999477, 999476, 999475, 999474, 999473, 999472, 999471, 999470, 999469, 999468, 999467, 999466, 999465, 999464, 999463, 999462, 999461, 999460, 999459, 999458, 999457, 999456, 999455, 999454, 999453, 999452, 999451, 999450, 999449, 999448, 999447, 999446, 999445, 999444, 999443, 999442, 999441, 999440, 999439, 999438, 999437, 999436, 999435, 999434, 999433, 999432, 999431, 999430, 999429, 999428, 999427, 999426, 999425, 999424, 999423, 999422, 999421, 999420, 999419, 999418, 999417, 999416, 999415, 999414, 999413, 999412, 999411, 999410, 999409, 999408, 999407, 999406, 999405, 999404, 999403, 999402, 999401, 999400, 999399, 999398, 999397, 999396, 999395, 999394, 999393, 999392, 999391, 999390, 999389, 999388, 999387, 999386, 999385, 999384, 999383, 999382, 999381, 999380, 999379, 999378, 999377, 999376, 999375, 999374, 999373, 999372, 999371, 999370, 999369, 999368, 999367, 999366, 999365, 999364, 999363, 999362, 999361, 999360, 999359, 999358, 999357, 999356, 999355, 999354, 999353, 999352, 999351, 999350, 999349, 999348, 999347, 999346, 999345, 999344, 999343, 999342, 999341, 999340, 999339, 999338, 999337, 999336, 999335, 999334, 999333, 999332, 999331, 999330, 999329, 999328, 999327, 999326, 999325, 999324, 999323, 999322, 999321, 999320, 999319, 999318, 999317, 999316, 999315, 999314, 999313, 999312, 999311, 999310, 999309, 999308, 999307, 999306, 999305, 999304, 999303, 999302, 999301, 999300, 999299, 999298, 999297, 999296, 999295, 999294, 999293, 999292, 999291, 999290, 999289, 999288, 999287, 999286, 999285, 999284, 999283, 999282, 999281, 999280, 999279, 999278, 999277, 999276, 999275, 999274, 999273, 999272, 999271, 999270, 999269, 999268, 999267, 999266, 999265, 999264, 999263, 999262, 999261, 999260, 999259, 999258, 999257, 999256, 999255, 999254, 999253, 999252, 999251, 999250, 999249, 999248, 999247, 999246, 999245, 999244, 999243, 999242, 999241, 999240, 999239, 999238, 999237, 999236, 999235, 999234, 999233, 999232, 999231, 999230, 999229, 999228, 999227, 999226, 999225, 999224, 999223, 999222, 999221, 999220, 999219, 999218, 999217, 999216, 999215, 999214, 999213, 999212, 999211, 999210, 999209, 999208, 999207, 999206, 999205, 999204, 999203, 999202, 999201, 999200, 999199, 999198, 999197, 999196, 999195, 999194, 999193, 999192, 999191, 999190, 999189, 999188, 999187, 999186, 999185, 999184, 999183, 999182, 999181, 999180, 999179, 999178, 999177, 999176, 999175, 999174, 999173, 999172, 999171, 999170, 999169, 999168, 999167, 999166, 999165, 999164, 999163, 999162, 999161, 999160, 999159, 999158, 999157, 999156, 999155, 999154, 999153, 999152, 999151, 999150, 999149, 999148, 999147, 999146, 999145, 999144, 999143, 999142, 999141, 999140, 999139, 999138, 999137, 999136, 999135, 999134, 999133, 999132, 999131, 999130, 999129, 999128, 999127, 999126, 999125, 999124, 999123, 999122, 999121, 999120, 999119, 999118, 999117, 999116, 999115, 999114, 999113, 999112, 999111, 999110, 999109, 999108, 999107, 999106, 999105, 999104, 999103, 999102, 999101, 999100, 999099, 999098, 999097, 999096, 999095, 999094, 999093, 999092, 999091, 999090, 999089, 999088, 999087, 999086, 999085, 999084, 999083, 999082, 999081, 999080, 999079, 999078, 999077, 999076, 999075, 999074, 999073, 999072, 999071, 999070, 999069, 999068, 999067, 999066, 999065, 999064, 999063, 999062, 999061, 999060, 999059, 999058, 999057, 999056, 999055, 999054, 999053, 999052, 999051, 999050, 999049, 999048, 999047, 999046, 999045, 999044, 999043, 999042, 999041, 999040, 999039, 999038, 999037, 999036, 999035, 999034, 999033, 999032, 999031, 999030, 999029, 999028, 999027, 999026, 999025, 999024, 999023, 999022, 999021, 999020, 999019, 999018, 999017, 999016, 999015, 999014, 999013, 999012, 999011, 999010, 999009, 999008, 999007, 999006, 999005, 999004, 999003, 999002, 999001 }

Returns: 1650.001627229141

Statistics

Problem Statement for "CasinoGame"

Problem Statement

Definition

Notes

Constraints

Examples