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« on: April 20, 2012, 03:49:55 am »
Here's my reasoning:
After you have dealt the first hand, you are in one of the following two situations:
(A) Eight cards left in the deck, including a Laboratory
(B) Seven cards left in the deck, not including a Laboratory
After you deal from A, you get to one of the following two situations:
(C) Three cards left in the deck, including a Laboratory
(D) Two cards left in the deck, not including a Laboratory
After you deal from (B), you get to (D).
After you deal from (C), you get to (B).
After you deal from (D), you get to either (A) or (B).
The probabilities of the interchange are:
A -> C with probability 3/8 and A->D with probability 5/8
B -> D with probability 1
C -> B with probability 1
D -> A with probability 8/11 and D->B with probability 3/11
The initial probabilities are A with probability 8/13 and B with probability 5/13.
... Maths stuff ...