How to calculate the probability when either one of two different cards will do

Question:

I wrote a program for calculating probabilities myself, but I am not sure how to develop unions. How big is P if I have

1 Lotus

4 Mana Vaults

4 Goblin Mutants

in a 60 cards deck and want to bring out the Goblin Mutant on 2nd turn. Of course I could turn the Lotus into a 5th Vault, but that is not very proper. I think.

Answer:

Let me see if I understand what you are asking. You want to know how likely it is to bring out a Goblin Mutant on turn two. Ordinarily, the only available mana sources at that time would be two lands (mountains), so you will need extra mana. You include both a Black Lotus and four Mana Vaults for this purpose. You are not asking for the likelihood of a single combination, but for the combined likelihood of two different combinations. One combination has the Goblin Mutant, land, and the Lotus. The other combination has the Goblin Mutant, land and a Mana Vault.

Assuming you have 20 mountains in your deck, there is a 7% chance that you will get the Goblin Mutant out with the Lotus and a mountain in nine cards. There is an 18% chance that you get the Goblin Mutant out with a Mana Vault and two mountains in nine cards (plus one point of mana burn). There is a 3% chance that you get both a Lotus and a Mana Vault in the first nine cards (along with the Goblin Mutant and two mountains). Therefore, there should be a 23% chance to get your Goblin Mutant out by the second turn. This number is obtained by adding the likelihoods of the alternatives taken separately, and then subtracting the likelihood that they happen together.

It makes sense to subtract the combined likelihood so that you do not double count those hands. The likelihood is derived from counting the number of hands that have the combination and dividing that count by the number of possible hands of a given size. Of the 960 million hands that have the Lotus combination, 400 million also have the Mana Vault combination. Similarly, of the 2.7 billion hands that have the Mana Vault combination, 400 million have the Lotus combination. So the number of hands that can cast the Goblin Mutant on turn two is 3.3 billion. (2.7 billion + 960 million - 400 million)

As you suspect, this is very close to what you would get if you just counted the Black Lotus as another Mana Vault. There are 3.1 billion hands that have the Mana Vault combination if you permit five of them in the deck. That does not take into account the fact that you need one less mountain when using the Black Lotus.

Using Deck-u-lator

For the Black Lotus combination, zero out the Mana Vault. Then get the likelihood for the first nine cards.

Black Lotus combination

have	need	card
		other
		Black Lotus
		Goblin Mutant
		Mana Vault
		Mountain

How many cards will you draw?

For the Mana Vault combination, zero out the Black Lotus and add another Mountain.

Mana Vault combination

have	need	card
		other
		Black Lotus
		Goblin Mutant
		Mana Vault
		Mountain

How many cards will you draw?

Add those first two likelihoods together and subtract the likelihood of the hand that has everything. You can get the likelihood for a hand with everything by setting the Black Lotus and the Mana Vault both to 1. (Make sure you include both Mountains.)

Black Lotus and Mana Vault combination

have	need	card
		other
		Black Lotus
		Goblin Mutant
		Mana Vault
		Mountain

How many cards will you draw?