Ex${{{}}}$tremely Tricky Probability Question? Here's the question. It's quite difficult: 
David is given a deck of 40 cards. There are 3 gold cards in the deck, 3 silver cards in the deck, 3 bronze cards in the deck and 3 black cards in the deck. If David draws the gold card on his first turn, he will win $50. (The object is to get at least one gold card). The other colored cards are used to help him get the gold card, while the remaining 28 do nothing.
David initially draws a hand of 6 cards, and will now try to draw a gold card, if he did not already already draw one. He may now use the other cards to help him. All of the differently colored cards may be used in the first turn.
David can use a silver card to draw 1 more card.
David can use a bronze card to draw 1 more card. However, he can only use 1 of these per turn.
David can use a black card to look at the top 3 cards of the deck, and add one of them to his hand. He then sends the rest back to the deck and shuffles. He can only use 1 of these cards per turn.
What are the odds David draws the gold card on his first turn?

We can ignore the silver cards-each should be replaced whenever we see it with another.  Similarly, if you have a bronze, you should draw immediately (but subsequent bronzes don't let you replace them).  So the deck is really $36$ cards, $3$ gold, $3$ black, and $30$ other (including the 2 bronzes after the first).  You win if there is a gold in the first $6$, or a black in the first six, no gold in 1-6 and a gold in 7-9, or a black in 1-6, another in 1-9, no gold in 1-9 and a gold in 10-12, or a black in 1-6, another in 1-9, another in 1-12, no gold in 1-12 and a gold in 13-15.  All these possibilities are disjoint, so you can just add them up.
