This is a rare recent appearance:
His attendance was perfect. He attended every possible meeting. And he sat mumchance throughout every meeting. I suppose it’s how you define work. You can sit like a lump throughout hundreds of meetings. Or you can engage your brain to question and to challenge.
Selkirk Weekend Advertiser, 11 Mar. 2010.
You may deduce that to remain mumchance is to stay silent, with a hint that to do so may be a sign of inferior intellect. In fact, in some English dialects its main sense has been remaining stupidly or solidly silent.
However, its first meaning was of a game of dice:
But, leaving cardes, lets go to dice a while,
To passage, treitrippe, hazarde, or mumchance.
Machivell’s Dog, an anonymous satire of 1617. The second of these games is frequently spelled trey-trip, because success in playing it depended on the casting of a trey, a three.
Nobody now seems to know the exact rules, though as it was often mentioned in the same breath as the dice game hazard, ancestor of craps, it’s assumed that it was similar.
The form and meaning of mumchance suggest it ought to be a close relative of mum in phrases like keep mum, stay silent. There is indeed a close connection, though the words have different origins. Mum is an imitative term known from the fourteenth century, while mumchance is sixteenth century, from Middle Dutch mommecanse, which has cousins in other Germanic languages and in French.
Paradoxically, the link between the dice game and silence is the notoriously noisy carnival, since it was traditionally played in the Netherlands during such festivities, in particular by mummers, masked actors in dumb-show. Mumchance was always played in silence, hence the sense.
The game of hazard, by the way, is the source of our word meaning a risk or danger. Mumchance seems to have been similarly perilous, as it evolved to mean a high-risk venture and continued in use in that sense after the dice game had been forgotten. The same name was also given to a card game, whose rules are as poorly recorded as those of the dice game, but which was also played in silence.