Claim: If you throw a die until the running total exceeds n>=5, a final outcome of n+1 is more likely than any other.

Assume we throw an m for a total n+k>n+1, and assume m-k>=0. Now, it is just as likely to throw an m as an m-k+1, which means that the sum n+1 is just as likely as any other. Now consider the series of throws consisting of n-5 1's followed by a 6 and note that we cannot achieve more than an n+1 by changing the last die roll. Hence, a total of n+1 is more likely than any other.

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