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As the discrepancy [from 1] in the sum of the nine probabilities seemed too blatant to be attributed to numerical error given the problem scale, I went and checked my R code for the probabilities and found a choose(9,3) instead of a choose(6,3) in the last line… The fit between the true distribution and the observed frequencies is now much better

but the chi-square test remains suspicious of the uniform assumption (or again of my programming abilities):

> chisq.test(obs,p=pdiag)
Chi-squared test for given probabilities
data: obs
X-squared = 16.378, df = 6, p-value = 0.01186

since a p-value of 1% is a bit in the far tail of the distribution.