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I disagree that there should be situations where the less likely situation is correct only becaus it is less likely ( as a pre-programmed result). The likelihood of an event occurring in the game should be a result of your acquired evidence and only 100% certainty can exist when there is enough concrete evidence supporting the outcome. Within the game it should be possible for the true outcome to receive a high probability. Your idea however is essential in situations where the probability of events are very close. For example in a situation with 5 outcomes where all their probabilities are 15-30% it wouldn't and shouldn't be obvious.
I think that lying should be possible from the beginning but, since you are a detective, you have the ability to gauge someone's reliability which is displayed as a percentage (like in your drawings). Also while reading I thought maybe it would be possible to combine 'evidence' to create new evidence. ie: Alice's shoes are wet && Bob's weather records show that there hasn't been rain in weeks +=Alice has stepped into the local lake for something today.
Can't we just assume that whatever we do was predicted correctly? The problem does assume an 'almost certain' predictor. Shouldn't that make two-boxing the worst move?