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I think your gut is right that there are not always clear ways to mathematize statistics problems, which is what accounts for debates about these kind of problems! I do think the calculation for a 17/18 chance in this problem is both satisfying and satisfactory. It is spot-on for gambling, the fair odds for making a bet, and it reflects mathematically what you say of becoming more convinced of our assumption. It can be very strange how knowledge changes probabilities... and it is not always clear how to recalculate based on the value of that knowledge!