You're on a game show and have a choice of three boxes, in one box is £10, 000 in the other two are nothing. You pick one box, the host then opens one of the other boxes showing it's empty, should you stick or switch?

At the moment of your first choice you have a 1/3 chance of picking the box with the prize and a 2/3 chance of getting nothing. After your choice those probabilities remain locked in and do not change as when you made that choice that was all the information you had.
At this point we can simplify the problems to two sets of probabilities, the box that you have chosen with a 1/3 chance of having the prize in it and the other two boxes that we now combine into one 'set' which together have a combined 2/3 chance of containing the prize.
After the host reveals one of the empty boxes (turning our artificial 'set' of two boxes into one real box) these probabilities remain the same. Meaning that sticking to your own box you remain having a 1/3 chance of winning but by switching you double your chances to 2/3.
This problem reveals a deep truth about probabilities that is hard to grasp, probabilities are always a reflection of the information that we have and never (until we start talking about quantum mechanics) a reflection of reality. This means that when the new information is revealed (the empty box being revealed) we must update our understanding of the probabilities of finding the prize in each of the two remaining boxes, not start again from scratch and think about two boxes one of which has a prize and therefore, incorrectly think that it is a 50/50 chance.