This applet allows you to consider many 12-card hands of Set. Prior to each hand, the deck is completely randomized. A 12-card hand is then dealt. If there is no set in the hand, this is recorded and the process begins again. If the hand has at least one set, one set is removed and the three cards replaced. Then this second hand is checked for existence of at least one set.

It appears that the fraction of first hands with sets is slightly greater than the fraction of second hands with sets. Question: Assume we know that the first 9 (of 12) cards dealt do not contain a set. What is the probability that the 12-card hand does not contain a set?