Problem #60
Rolling Dice
Alex has a 7-sided regular dice. Every side has a distinct value between 1 to 7, each with equal probability of coming up in a roll. Alex rolls the dice once. Let the value obtained be equal to S_{1}. Now he rolls the dice S_{1} number of times. Let the sum of values obtained in these S_{1} rolls be equal to S_{2}. Now he rolls the dice S_{2} number of times. Let the sum of the values obtained in these S_{2} dice rolls be equal to S_{3}. This process is continued for an infinite number of times. Find the expected value of S_{1182014}. If the answer is x, give your answer as \lfloor x \rfloor\mod 10^9+7, where \lfloor x \rfloor denotes the greatest integer less than or equal to x.