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 Quote by Simon 6 Ok, we've established that the odds are 50/50 if the Host doesn't know and 2/3 in favour of swapping if the Host does know. Lets see if you're ready for this: You select one of three cards, as before. Enter Host 1 who does not know where the Queen is. He randomly reveals a Jack. He then turns it back over and leaves. Nevertheless, you remember it is a Jack. Enter Host 2 who did not witness the above. He knows where the Queen is and reveals a Jack. You are shown nothing new. He has turned over the same card as Host 1. A game is cancelled if Host 1 reveals the Queen or if Host 2 reveals a different card. For all games that proceed: are the odds 50/50 or 2/3 in favour of swapping?
The same 50/50 analysis seems to apply. (Though I've been wrong before! )

Assuming I understand the rules this time:

1/3 of the time your initial pick is correct; Host 1 always a Jack; you lose when you switch.

2/3 of the time your initial pick is incorrect, but half of those times don't count since Host 1 shows the Queen; for the half that do count (2/3 * 1/2 = 1/3 of the initial starts), Host 2's actions change nothing and you win when you switch.

So, it's fifty-fifty!