[aamar]

If there was no guarantee that a number would appear 1.9 times, then seeing singleton numbers wouldn't be a predictor.

Let's say there's no such demand, and the boards are generated truly randomly and neutrally. If visible-quality X is disproportionately correlated with invisible-quality "Win" then the game is already flawed; this may emerge naturally from the Win conditions and from the game design decision of what is shown in the baited-hook. If the output is controlled post-generation to increase the proportion of Win cards and Near-Win cards vs. Lose and Invalid cards, then the statistical correlation may be greatly increased.

Simplified example: Scratcher with two numbers 0-4: one bait, one hidden, pays if sum is 5 or 6. If everything is fair and truly random, the odds of winning are, 0 showing: 0%, 1 showing: 20%, 2-4 showing: 40%. Already a bad game, but now the game designers want to eliminate cards with sums of 7 or 8 because this confuses people, (not minding that they're changing overall odds of winning), so they block those Invalid cards from shipping without blocking anything else. That gives:

    0 1 2 3 4
  0 - - - - - 
  1 - - - - W
  2 - - - W W
  3 - - W W I
  4 - W W I I

The new odds of a given ticket, given the visible number, are: 0: 0%, 1: 20%, 2: 40%, 3: 50%, 4: 67%. The point here is that now a 4-showing-card is >3x as good as a 1-showing card, when it used to be only 2x as good, and it might now have positive expected return.