No matter how you feel about the color of the sky, you are perfectly free to accept the bet and win if the sky is blue.
"perfectly free" basically supposes that you have free will. In practice human's like to believe that they have free will but they don't behave that way in experimental settings. As long as you think about thought experiments with your usual intuition that presupposes free will you don't get to the substance of the argument and understand how beliefs work in practice.
You can argue that we don't have direct write access to anything; but if you want to describe the facts with "we have direct write access to some things," then it is reasonable to include most aspects of our beliefs in that statement.
In this video, Douglas Crockford (JavaScript MASTER) says:
1
I don't think he has "absolutely no evidence". In worlds where DOUGLAS CROCKFORD has a gut feeling about something related to programming, how often does that gut feeling end up being correct? Probably a lot more than 50% of the time. So according to Bayes, his gut feeling is definitely evidence.
The problem isn't that he lacks evidence. It's that he lacks communicable evidence. He can't say "I believe A because X, Y and Z." The best he could do is say, "just trust me, I have a feeling about this".
Well, "just trust me, I have a feeling about this" does qualify as evidence if you have a good track record, but my point is that he can't communicate the rest of the evidence his brain used to produce the resulting belief.
2
How do you handle a situation where you're having a conversation with someone and they say, "I can't explain why I believe X; I just do."
Well, as far as updating beliefs, I think the best you could do is update on the track record of the person. I don't see any way around it. For example, you should update your beliefs when you hear Douglas Crockford say that he has a gut feeling about something related to programming. But I don't see how you could do any further updating of your beliefs. You can't actually see the evidence he used, so you can't use it to update your beliefs. If you do, the Bayes Police will come find you.
Perhaps it's also worth trying to dig the evidence out of the other persons subconscious.
3
Ok, now let's talk about what you shouldn't do. You shouldn't say, "Well if you can't provide any evidence, you shouldn't believe what you do." The problem with that statement is that it assumes that the person has "no evidence". This was addressed in Section 1. It's akin to saying, "Well Douglas Crockford, you're telling me that you believe X and you have a fantastic track record, but I don't know anything about why you believe it, so I'm not going to update my beliefs at all, and you shouldn't either."
Brains are weird and fantastic thingys. They process information and produce outputs in the form of beliefs (amongst other things). Sometimes they're nice and they say, "Ok Adam - here is what you believe, and here is why you believe it". Other times they're not so nice and the conversation goes like this:
Just because brains could be mean doesn't mean they should be discounted.