Can someone do a simulation? I don't believe the probability of winning could be doubled by changing the choice.
However, I must admit to finding the cited problems, in pure math, neither practical nor interesting. There was one old brain-teaser I found appealing and remembered for a little talk to a non-mathematical group. A well-known tv hostess with a super IQ caused a storm of controversy with it:
Suppose you are on a quiz show. You may choose to open one of three doors. One has a prize behind it, an other a booby prize. Suppose you choose one door but before you open it, the show hostess, opens another door which reveals no prize. The hostess then allows you to stay with your first choice or to choose, instead, the other unopened door.
The question is: which is the best strategy?
The tv woman with the genius IQ said: change your choice. Letters at a rate of nine to one disagreed with her, including some academics, on the degenerate influence of tv, saying things like: you really blew it, this time! Their argument was that the move from one door to the other shouldnt make any difference, because there was an equal probability that the prize would be behind either still unopened door.
Of the little group, this reviewer talked to, some guessed right some wrong. None really know. I had thought like the ignorant ninety per cent. The interesting thing is that the most prolific mathematician of the twentieth century couldnt understand, either. Like a green student, he pestered his host and colleague for an explanation.
Erdös was shown a computer simulation of the quiz show given a large number of trials. On average, the probability of winning the prize was one-third, if one stayed at the door of one's first choice. If one changed one's choice, the probability of winning became two-thirds.
Erdös accepted the result but he still wanted a transparent explanation 'straight from the book'.
Erdös' friend and colleague put it this way. You, as the quiz contestant, know you are going to be given the chance of making two choices for the prize.