How many elves to have Priest of Titania reliably tap for 2+ mana?

[The Golgari Guy]

I was taking a look at my green decks, and noticed that in many of them I incidentally run so many elves (examples: Llanowar Elves, Deathrite Shaman, Reclamation Sage, Beast Whisperer, Springbloom Druid...) that I can probably include a Priest of Titania and have it tap for 2+ mana more or less reliably.

However, I would like to calculate (maybe even just an estimate) how many elves I need to have, let's say, a 90% chance of tapping Priest for 2+ mana on turn 3 (assuming I cast it on turn 2).

Can someone help me out?

[Morganator 2.0]

Do you know what my favorite thing about Deckstats is? The stats.

I answered this already in a previous thread, but it was with Mox Opal. Fortunately, because Mox Opal needs 2 other artifacts to work, and you're looking at the odds of having two other elves, the math works the same.

https://deckstats.net/forum/index.php/topic,52447.msg151035.html#msg151035

Short answer: about 30 elves. If your commander is an elf this makes things easier. This is what you wanted right?

[The Golgari Guy]

Do you know what my favorite thing about Deckstats is? The stats.

I answered this already in a previous thread, but it was with Mox Opal. Fortunately, because Mox Opal needs 2 other artifacts to work, and you're looking at the odds of having two other elves, the math works the same.

https://deckstats.net/forum/index.php/topic,52447.msg151035.html#msg151035

Short answer: about 30 elves. If your commander is an elf this makes things easier. This is what you wanted right?

Thanks for the reply!

Since the Priest counts herself I would only need (at least) another elf for her to tap for (at least) 2 mana. The probability that it taps for at least 2 is what I'm interested in. For simplicity, we can assume that the Priest is in my opening hand. I guess in that case I'd need way fewer than 30?

I would also be interested in the reasoning and/or the formulas used to obtain those probabilities, if you'd like to share them.

[The Golgari Guy]

Ok, so I tried to make the question more precise and then to work out the answer using an hypergeometric calculator (https://aetherhub.com/Apps/HyperGeometric). Maybe you can tell me if the math checks out.

Let's assume that Priest of Titania is in my opening hand, and that I have N other Elves in my deck (for example we can assume N=10). I'll also assume that all these Elves have CMC=3 or less, so that I can play them on turn 3.

I play Priest of Titania on turn 2. What is the probability P that I will able to tap her for more than 1 mana during turn 3? Or, in other words, what is the probability that I draw at least another Elf before turn 4?

Related question: How large must be N to make P=0.90?

So, my reasoning is the following: I assume one of the cards in my opening hand is the Priest, and in addition I have my commander in the command zone. So my sample size is 98 cards.

On T1, I see 8 of these cards, of which 1 is the Priest. So, I see 7 of them. If N=10, the probability that one of those is an Elf should be P=54.1%. On T2, I will have seen 8 total, etc., so that:

T1: P=54.1% (cards seen: 7+1)
T2: P=59.2% (cards seen: 8+1)
T3: P=63.7% (cards seen: 9+1)

This should be the probability to tap the Priest for more than 1 on turn 3.

I've also made some trials and found that I need 21 other Elves with CMC less than or equal to 3 to have P=0.90 (0.897 to be precise).

Is this correct?

[Morganator 2.0]

If you're looking to only have one other elf in addition to Priest of Titania, then yes, your odds are at 89.7% of achieving this on turn 3 with 21 other elves in the library. Looks like I misunderstood things the first time. I though you wanted two other elves, and missed the 90% success chance.

[Slyvester12]

In this deck:
https://deckstats.net/decks/132279/1577597-ezuri-elfball-combo-primer
I only run 20 elves, I believe. I never run into problems with any elves-matter effects.