Your probability of the £250k being in the box, is exactly the same, at all points in the game, as £1, £5 or any other uneliminated amount being in the box.
Trazor's post a couple above this explains it better - The odds of it being on one side of the fence or the other reduce over time, but at the end, its a pure 50/50...
EDIT: Or, put another way, at the beginning of the game, the chances of £1 being your box are the same as the chances of £250,000 being in your box. What do you propose happens during the game to change those odds ?
Let's try it another way. Take a game with four boxes, A, B, C and D:
Box A holds £100
Box B holds £1
Box C empty
Box D empty
I don't know which box is which.
I have to choose one box, which then remains unopened until the end of the game. Two of the remaining boxes are eliminated and I can then stick with my first choice or switch to the other remaining box. Obviously there are four possible choices for my first box so we will look at each of the four possible scenarios. In each case, box A remains as one of the last two boxes with one of either B, C or D. I adopt a strategy that I will always switch, so open the other remaining box. Let's work through the possibilities.
A1. I choose box A
A2. (say) C & D are eliminated. I switch to B and lose
B1. I choose box B
B2. C & D are eliminated. I switch to A and win
C1. I choose box C
C2. B & D are eliminated. I switch to A and win
D1. I choose box D
D2. B & C are eliminated. I switch to A and win
Four scanarios. I win 3 and lose one so my switch strategy gives me a 3 in 4 chance of success once the game has come down to a choice of two boxes and the £100 prize is still in the game
This is a parallel situation (with fewer starting boxes) described in the OP except that the OP stipulates that, in that particular case, the £1 box (box B in my example) remained as well as the top prize. Now it's true that if you stipulate that box B and only box B
can be the second remaining box, then the chance is 50:50, but that artificially forces B to remain in the game, which destroys the essence of the problem. In practice, I don't care whether my final choice is between A and B, A and C, or A and D. In the real game situation, the switch strategy gives the advantage.