You and your two friends are playing a game where you each draw a random real number, \(r_i\), between 0 and 1. Call this your score. You win the game if you draw the largest score of all: \[\max \{r_1,r_2,r_3\}\]

In two of every three games, you lose the game. However, on the occasions you do win you tend to go big.

Assume that you and your two friends play this game long enough to collect accurate statistics. Out of all the occasions in which you win the game, what is your average score?

**Assumptions**

- There is no betting component to this problem.

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