None of this sits well with me. That tiny range after 100 trials is too low. This says that you expect to have exactly $0 after 100 trials about 19 times out of every 20 experiments. Gut check says no.

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Isn't StDev = sqrt(2) units per trial? So StDev after 100 trials is 141.4?

141.4 /sqrt(100) = 14.14
14.14*1.92 = 27.15

Using 1.96 is for "infinite trials", and generally is a fine approx for more than 20 trials, but since 1.92 is technically correct here, I'm using it.

(I start work in 15 minutes, so I can't really get into this right now. I'll think on it this afternoon.)