It later turned out that the two were childhood acquaintances who were not romantically involved\. What went wrong?
I find the entire explanation described below very misleading and perhaps even largely incorrect. The workshop participants had it wrong mostly for two reasons:
They did not consider what is the likelihood of visiting a museum / workplace given any other alternative (mutually exclusive) relationship - not strangers but also not romantically involved; i.e, friends. Being acquaintances is not a relevant type of a relationship as it is not mutually exclusive with a romantic relationship (a pair can be both dating and working together).
They did not know the prior probability of an arbitrary pair of people being romantically involved. A naive assumption of 50% of them being romantically involved is wrong, and should be made by observing the proportions of romantic relationships in the population.
In terms of the previous coins-fairness example, they (a) only considered that one type of coin (fair) is 2 times as likely to turn up heads as another type of coin (tail-biased), but did not consider how likely are the other type of coins (head-biased) to turn up heads; and (b) they did not know the proportions of coin types in the bathtub.
The explanation below also fails to mention the important assumption that the trait being assessed in all of the examples (coins, emails, workshop) is constant and doesn't change over time. It is important to mention because it may not be so trivial for every example, yet it reduces the complexity of the estimations tremendously. A coin is not expected to change its bias significantly over time, yet a relationship does, and so does the magnitude of "spamness" in a given mail for a given person (for instance, when I get older I may be more interested in pharmaceutical ads).