Monty Hall revisited

Logic Level 5

Suppose you're a contestant on a game show and the host shows you 5 curtains. The host informs you that behind 3 of the curtains lie lumps of coal, but behind the other 2 curtains lie separate halves of the same $10,000 bill. He then asks you to choose 2 curtains; if these 2 curtains are the ones with the 2 bill halves behind them then you win the $10,000, otherwise you go home with nothing.

After you choose your 2 curtains, the host opens one of the remaining 3 curtains that he knows has just a lump of coal behind it. He then gives you the following options:

(0) stick with the curtains you initially chose,
(1) swap either one of your curtains for one of the remaining curtains, or
(2) swap both of your curtains for the remaining 2 curtains.

Let \(p(k), k = 0, 1, 2\), be the respective probabilities of winning the money in scenarios \((k)\) as outlined above. Then

\[p(2) - p(0) - p(1) = \dfrac{a}{b},\]

where \(a\) and \(b\) are coprime positive integers. Find \(a + b\).


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