Probability in a card game

I play a card game called 'Yu-GiOh!'. For those who don't know how this game works, that will be separate paragraphs at the bottom of the page.

I have a deck, and I'm looking for the chances of getting a specific one of these using only the starting hand. The numbers are below, (a) and (b) etc. denotate that the cards mentioned are different than other mention cards. For the math I'm assuming the opponent cannot negate anything.

Total deck size: 40
Starting hand: 5 cards

Cards: 1-3: The key card
Cards 4-6: Can search the key card if [29] isn't in the hand.
Cards 7-9: Can search the key card if [29] isn't in the hand.
Cards 10-12: Can search the key card.
Cards 13-15: Can search the key card if one of [22-28, 29] or [7-9] are in the hand.
Cards 16-18: Can search the key card if [29] is in the hand
Cards 19-21: Allows me to draw 2 cards. If used, another copy cannot be used)
Cards 22-28: Cards that are useless by themselves.
Card 29: A card useless by itself.
Cards 30-40 || Not useful to search the key card.
For more details to what each card does, message me.

The following paragraphs explain how the game works.

You start a game with 5 cards in your 'hand'. You draw these 5 starting cards from a 'deck' which can have between 40 and 60 cards (inclusive).
Other than on the first turn, you draw 1 card before anything else. Using your hand, you can put cards on the 'field' to use them, either by summoning them (monsters), activating them (spells), or setting them facedown. Some cards can only be used on some of these ways, and some cards ignore them. Using cards on your field, there are a few ways to win.

The main way to win is to reduce your 'opponent's life points', an HP bar, to 0. Monsters attacking can do this, and some cards also affect this number.
The secondary way is to have a card that has an effect to win the game. For instance the card "Holactie, the creator of light". If he's summoned, you win.
The third way to win is when your opponent tries to draw a card, but can't. There are cards that add or remove cards from the deck.

This is a barebones explanation, but it should explain enough.