Without skill or transaction costs, the distribution of terminal one-year wealth (cumulative trading profits) would have a near zero mean but would be heavily right skewed. The median and mode would be to the left of (less than) the mean. So more than 50% would be losers. The math of this was worked out in a Bouchaud paper around six or eight years ago and is also touched on in Piketty's book a few years ago (and expanded on a little more in a recent paper where Piketty was a co-author, the name of which escapes me at the moment). I'll try to find the Bouchaud paper and post it here tomorrow if I find it.
Edit: The argument in Piketty et al is obscured by their focus on generally positive return on capital (postive interest rate, investment return, rent, ect) but it works out even if expected return is zero. Consider two consecutive coin flips, risking 100% of your stake on each flip. Expectation is zero, but 75% of the time you'll end up minus 1 and 25% of the time +3. A bit like the Petersburg game: expectation (mean) is infinity but you'd only pay a few dollars for it.