Which, in case anybody's forgotten by now, was what they needed to prove that:

1/INF * [ (0/INF)^N + (1/INF)^N + (2/INF)^N + ... + (INF/INF)^N ]  =  1/(N+1)

...which in turn is how you represent an infinite number of possible hypotheses about how far from 0% LEFT to 100% LEFT the first Ball 0 could have landed, each hypothesis with equal prior probability 1/infinity, and then update on seeing N balls go LEFT, as would have a probability of p^N for each hypothesis a fraction p of the way between the left side and right side.

Also in case anybody's forgotten, if you imagine N balls plus the 0 ball getting randomly ordered, the chance of the 0 ball coming rightmost in the ordering - hence, all other balls being LEFT of it - is 1/(N+1), which is how they originally knew that was the desired end result.

Which, in case anybody's forgotten by now, was what they needed to prove that:

1/INF * [ (0/INF)^N + (1/INF)^N + (2/INF)^N + ... + (INF/INF)^N ]  =  1/(N+1)

...which in turn is how you:

- Represent an infinite number of possible hypotheses, about how far from 0% LEFT to 100% LEFT the first Ball 0 could have landed;
- Each hypothesis with equal prior probability 1/infinity;
- And then update on seeing N balls go LEFT;
- As would have a likelihood of p^N for each hypothesis a fraction p of the way between the left side and right side.

Also in case anybody's forgotten, if you imagine N balls plus the 0 ball getting randomly ordered, the chance of the 0 ball coming rightmost in the ordering - hence, all other balls being LEFT of it - is 1/(N+1).  Which is how they originally knew that 1/(N+1) was the desired end result, for the likelihood of seeing all N balls land on the LEFT.