Willy Plays Craps Roller

Willy goes to Las Vegas and plays the game Craps Roller.

The rules of the game Craps Roller are as follows:

One player rolls two die. If the sum of the numbers of the two die is equal to 7 or 11, the player wins. If the sum of the numbers is 2, 3, or 12, then the player loses. If the sum of the numbers is equal to something else, say \(x\), the die are rolled again until either the sum of the numbers is \(x\) or the sum of the numbers is 7. If \(x\) is rolled before 7, then the player wins; otherwise, the player loses.

Willy becomes addicted to Craps Roller and plays it 990 times. What is the expected value of the number of times he will win? Round your answer to the nearest integer.

(Dispute: If the player rolls a sum that is, let's say, 4, he keeps rolling until he gets either 4 or 7 - any other valid numbers from the first part will not be considered.)

(Arman Siddique: Programming is not allowed.)


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