I'm answering questions from the 'hardest exam in the world'

Upvote, because I didn't know the 'hardes exam in the world' existed, and I now think it's awesome. Good luck.

On "Intuition 2":

> At time t, if it is true that the probability of the marble being under a given box is 0.5, then if I guessed randomly, I would be correct one out of two times on average.

> The fact that the probability is 0.5 is completely independent of whether or not any particular mind believes that it is 0.5. But this is the definition of an objective truth. Therefore, probability must be objective.

This is begging the question. The two things combined is like saying "At time t, if it is true that probability is objective... therefore probability must be objective."

I understand the question quite differently. The question of objective probabilities is a question of whether the world is deterministic.

If you believe the world is governed by physical laws and you further believe you understand those laws (physics), you consequently must believe that with sufficient information about the past you can predict the future (determinism). So if I was to throw a dart at a dartboard, with sufficient information about arm speed, humidity, wind etc you could perfectly predict where on the dartboard it would land. The problem is that the mathematical / computational complexity necessary to achieve that is still beyond our means. Enter Probability, which we use to reduce the computational burden of predicting the future. As probability will always be model dependant (ie subject to someone's view of the world and necessarily wrong some % of the time) - it follows that an objective probability cannot be consistent with a deterministic world.

So is the world deterministic?

Saying p=0.5 necessarily implies the 1/2 thing. Neither implies any stance on whether it's subjective/objective so far. The 'objective' conclusion there requires the further fact of mind-independence, which is based on 'you can't change the odds that a fair coin comes up heads just by thinking about it' type arguments. So it's not circular/question-begging, IMO.

When a fair coin is tossed it will either come up heads or it wont. Put another way, the probability of getting heads is 1 or 0, unless you believe that something magical happens to the coin while it is in mid-air that somehow makes it impossible to predict what will happen when it lands (given a complete set of data).

So the outcome of the coin-toss is knowable, it just happens to be complex enough that it is not known (to us). To bridge the gap between the knowable and the not known we introduce a margin of error in our calculations, called Probability. Probability then, only exists as man-made mathematical spackle - and is by definition subjective.

To prove me wrong find an event that has an outcome that cannot be known in advance. That's why this question is on a philosophy exam.