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As far as I've noticed, as long as you haven't exhausted your supply of native heroes in your employ, the chance of a native hero appearing in the tavern at the beginning of the week isn't 1/9, but rather approximately (1 - 1/9), and the chance of a second native hero appearing is 1/9. This means that at the beginning of the week, the appearance of one native hero is almost guaranteed. In all other cases, the chance will be 1/9.
