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# Expected number of dice rolls

Due: January 19
Points: 2

The previous problem talks about a common way to choose who goes first among $$n$$ players of a game:

1. If $$n=1$$, there is just one player remaining so they go first.
2. Otherwise, if $$n\ge 2$$, each player rolls 2 dice and takes the sum.
3. Any players who do not have the highest sum drop out; they won't be going first.
4. Set $$n$$ to the number of players who tied for the highest, and return to Step 1.

For $$n=1$$ up to $$n=10$$, calculate the expected number of total dice rolls that $$n$$ players will make in order to decide who goes first. For example, with $$n=3$$, you know the expected number is at least 6 because every player starts out by rolling 2 dice. But if there's a tie, there's a chance that more rolls will happen. So the expected value for $$n=3$$ will be somewhat greater than 6.

Hint: In fact, the expected number of rolls when $$n=3$$ is $\frac{3888}{575} \approx 6.76174.$

(You can come up with a formula for this if you want, but I only need the values for $$n=1$$ up to $$n=10$$, accurate to at least 4 decimal places.)