There's actually an easier way to remove the subtleties of this statistical problem by using a hyperbolic example. For kicks and shiggles we'll up the stakes to

There's actually an easier way to remove the subtleties of this statistical problem by using a hyperbolic example. For kicks and shiggles we'll up the stakes to be a prize of $1B USD. Consider instead a Monte Hall scenario with 100 boxes (vice just 3), maybe we call this the "deal or no deal" variant of the problem... The user picks 1 box and has a 1/100 chance of selecting the box with the prize. Now, the host opens 98 boxes that they know do not have keys in them, leaving two unopened boxes (the one the user picked and the one the host left unopened). Now, pick your box from the remaining selection and claim your prize. You bet your sweet bippy I know which box I'm picking. Information is gained by observing the winnowing of the field of options.