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when playing the game one is generally attempting to achieve a particular double

Perhaps, but if that's what you meant it might be worth being clearer about that; the way it's phrased aþm suggests that the probability of being able to make any Séj‐dice capturing move (assuming availability of pieces to capture) is 1∕3+1∕36=13∕36, which doesn't really make sense (not least, that'd be likelier than merely being able to move even if the 1∕3 figure were correct). The actual probability is in fact, as dax00 said, 11∕36(=chance of matching the last piece to move)×1∕6(=chance of a double)=11∕216. The chance of any given piece being able to capture is 1∕6 of that again, i.e. 11∕1296.

since the fraction 11∕36 is almost a third, doesn't that in effect equate to 1∕3?

Well 1∕3=12∕36, so… no? It's close, sure, but still an 8.33% difference — if you consider that trivial enough to be discounted fine, but don't expect everyone (especially those of us with a mathematical inclination) to agree.

There is a 66.6% chance that, because each turn the dice MUST match the opponent's piece, thence the game will continue as regular Classical Chess

I also just noticed this remark; even aside from the percentage being wrong — the chance that a given turn will be ‘normal’ is 25∕36=69.44% — it's not clear whether you mean that to apply only to each turn, or (incorrectly) to the whole game. After 2 turns the likelihood of still having a normal game is 25∕36×25∕36=625∕1296 (less than 1∕2) and it keeps going down from there. Having a full game of Séj where the dice do not once allow a deviation from ‘classical’ chess is vanishingly unlikely.