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Expected Value of Modified Dice

Jane Street● MediumINTERNONSITE INTERVIEW

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Problem statement

Ten fair six-faced dice are supplied by their complete integer face lists. Die i is rolled rollCounts[i] times, and the counts sum to exactly 100. The game pays the sum of all rolls and charges a fixed entry fee of 350.

Return eleven reduced fractions in numerator/denominator form. The first ten values are the expected face value of each die in index order. The final value is the expected net payout after subtracting the entry fee. A zero value is returned as 0/1, and denominators are always positive.

Function

modifiedDiceExpectations(dice: int[][], rollCounts: int[]) → String[]

Examples

Example 1

dice = [[1,2,3,4,5,6],[1,2,3,4,5,6],[1,2,3,4,5,6],[1,2,3,4,5,6],[1,2,3,4,5,6],[1,2,3,4,5,6],[1,2,3,4,5,6],[1,2,3,4,5,6],[1,2,3,4,5,6],[1,2,3,4,5,6]]rollCounts = [10,10,10,10,10,10,10,10,10,10]return = ["7/2","7/2","7/2","7/2","7/2","7/2","7/2","7/2","7/2","7/2","0/1"]

Every die has expectation 3.5, so one hundred rolls have expected payout 350 and net expectation zero.

Example 2

dice = [[1,1,3,4,5,6],[1,2,2,4,5,6],[1,2,3,3,5,6],[1,2,3,4,4,6],[1,2,3,4,5,5],[2,2,3,4,5,6],[1,3,3,4,5,6],[1,2,4,4,5,6],[1,2,3,5,5,6],[1,2,3,4,6,6]]rollCounts = [100,0,0,0,0,0,0,0,0,0]return = ["10/3","10/3","10/3","10/3","10/3","11/3","11/3","11/3","11/3","11/3","-50/3"]

Only the first modified die is rolled, giving expected payout 1000/3 and net expectation -50/3.

Constraints

dice.length == rollCounts.length == 10.
Every die has exactly six integer face values.
0 <= rollCounts[i] <= 100 and their sum is 100.
Every face value is between -10^6 and 10^6.