> finding 100$, you do not know if it is the low or high amount.

Indeed, it's a 50/50 chance whether you pick the high or low amount.

> The computation is NOT $50*1/2+ $200*1/2 ( = $125)
> but either :
>
> $50*1/2 + $100*1/2 = $75
> or
> $100*1/2 + $200*1/2 = $150
>
> Which turn out $75*1/2 + $150*1/2 around $112.5... if that was random.

So you'd always change and take the other box?  Why even bother opening the 
1st box, just leave it shut and take the 2nd one.  Oh but hang on, then when 
you look inside the 2nd one you might want to change back to the 1st...

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