The tavern IS rigged. Look at this statement from the wiki

You would think that the odds of obtaining a card in the shown set of 5 is 1 in 5, however this is not true. The deck is actually just an illusion and the card(up to 3) that you will actually receive (or not receive) is predetermined. The odds of flipping over a hero when it is in the deck you play is probably on par with obtaining a gold rune from the rune temple. It seems to be around 1.25% - 2.5% chance of flipping over the gold card.