More Dice, More Problems

Suppose Alice comes across an infinite collection of dice that are $n-$ sided with values $1,2,\dots, n$ on the sides. The dice are all biased the same way i.e. side $i$ shows up with probability $p_i$ . For some positive integer threshold $s$ , Alice and Bob play the following game, with Alice starting: Alice rolls $1$ die from the collection. Call her roll $T_1$ . If $T_1 > s$ , Alice wins the game. Otherwise, Bob rolls $T_1$ dice from this collection. Call the sum of his $T_1$ rolls $T_2$ . If $T_2 > s$ , Bob wins the game. Otherwise, Alice rolls $T_2$ dice from the collection. Call the sum of her $T_2$ rolls $T_3$ . If $T_3 > s$ , Alice wins the game, otherwise, repeat in the same fashion as the above until someone rolls a sum of dice greater than $s$ . Find Alice's probability of winning the game in the case where $s = 2, n = 6$ , and the dice are fair. Note that unlike other problems, there is nowhere near a closed-form expression for this probability for general dice and values of $s$ .

Answer format: round to 3 decimal places.