Hearthpwn

I continually seem to be correcting people in threads about the chances to draw a certain card by X point in the game. This isn't as simple of a calculation as most people seem to think it is. From my understanding of how it works, here is the table that outlines your chances to draw at least one of two copies of a certain card in your deck, assuming you go first (start with 3 cards) and mulligan aggressively for it (throw everything else back).

It is a hypergeometric distribution complicated by the fact that the deck "resets" after the mulligan phase. In this model, you have a 41.3% chance of starting with the card you want by the time you play your first turn. Your chance of having a Wild Growth by turn 2 is 45.8%.

The same numbers for going second are 46.9% and 50.1% respectively. 

wow, thanks for sharing...... this is good to know.

I like what you have here, this needs to exist. Some cards have 1 copy, so I'd like to see a version that covers both if 1 copy exists and if 2 copies exist and have it all in one place. I'll have to sit down with my python idle and do some math at some point and post it up sometime.

This is a good table it's helpful to see the numbers. My gut reaction, without doing any calculation, is that your mulligan numbers are wrong - for the reason you allude to in your final paragraph. You can redraw the cards your threw away in the mulligan phase. Is that being factored in appropriately? I know this to be a possibility because I just threw away a Molten Giant in Arena (my only one) and took it back in the mulligan phase.

Quote from Mister_Smith »

I'm really bad at stuff like this so i will leave potential errors in the math for other guys to find but if i read that correctly it looks like your table is saying you've got 28 cards left after the initial draw... buuut that doesn't make sense?

What it is saying is that 28 of the 30 cards you can draw on your first draw and not the card you are looking for. Until the reset after the mulligan phase it is just a "simple" hypergeometric, you can check the math here

And looking more carefully, it still feels wrong to me. I know statistics can sometimes be non-intuitve. When you have 3 cards left, any given card that you haven't seen yet is at most 67% of your remaining deck. I'd think your chances of drawing it are 67%, not 99.8. Am I wrong?

Quote from jfmario »

I like what you have here, this needs to exist. Some cards have 1 copy, so I'd like to see a version that covers both if 1 copy exists and if 2 copies exist and have it all in one place. I'll have to sit down with my python idle and do some math at some point and post it up sometime.

This is a good table it's helpful to see the numbers. My gut reaction, without doing any calculation, is that your mulligan numbers are wrong - for the reason you allude to in your final paragraph. You can redraw the cards your threw away in the mulligan phase. Is that being factored in appropriately? I know this to be a possibility because I just threw away a Molten Giant in Arena (my only one) and took it back in the mulligan phase.

Actually I remember Ben Brode confirming somewhere that in constructed at least you cannot get back the cards you mulligan. If someone has evidence otherwise I am happy to fix this but I specifically remember him saying that in an interview

I'm very possibly mistaken. However, I still think the numbers are wrong, which is most evident to me when you give a 99.8% chance to draw the card when you have 3 cards left. Unless the system works differently than I think it does, this is not right. I'd like to hear your logic on this as perhaps my intuition is wrong and I'd like to know whatever it is I'm missing.

As requested here is the table for a card with only 1 copy in your deck. Sometime when I'm bored I am will do the table for chance to draw a 2-part combo like FoN + Savage Roar

to explain where I'm coming from better, I think the chance to draw a card in hearthstone is the number of copies of the card divided by the number of cards remaining in your deck - and nothing else. I don't see why a hypergeometric distriubtion is the way to represent this.

Now that I see this applied to 1 copy of a card I am more confident this is wrong. When you haven't seen your one copy of a card and you have 2 cards in deck, the chance that that card is on top is exactly 50%.

Edit:  I think I see what I wasn't getting. My bad. These are the numbers from the beginning of the game. 97% chance that you'll draw it by draw 26, is that it? I was thinking of it resetting everytime if you haven't drawn it yet - but this table appears to just be the outlook as of the beginning of a game. 

Quote from jfmario »

I'm very possibly mistaken. However, I still think the numbers are wrong, which is most evident to me when you give a 99.8% chance to draw the card when you have 3 cards left. Unless the system works differently than I think it does, this is not right. I'd like to hear your logic on this as perhaps my intuition is wrong and I'd like to know whatever it is I'm missing.

You misunderstand. This is not the chance to draw the card on that turn, this is the chance to have drawn the card by that turn. The 99.8% is correct as far as I can tell. 

Here is a Blizzard employee confirming the discarded cards are set aside and cannot be drawn again in the mulligan phase

Ok, got it. As long as people looking at this understand that if they reach 3 cards left and haven't seen it, their chance of getting it then is 67 and not 99. I was interpreting what this card was trying to do incorrectly. Which is why it looked correct to me at the top but the bottom looked wrong.

Quote from Semioteric »

Here is a Blizzard employee confirming the discarded cards are set aside and cannot be drawn again in the mulligan phase

Great. Thanks for the link, I guess I had 2 copies then and thought I had one.

Quote from jfmario »

Now that I see this applied to 1 copy of a card I am more confident this is wrong. When you haven't seen your one copy of a card and you have 2 cards in deck, the chance that that card is on top is exactly 50%.

 

Edit:  I think I see what I wasn't getting. My bad. These are the numbers from the beginning of the game. 97% chance that you'll draw it by draw 26, is that it? I was thinking of it resetting everytime if you haven't drawn it yet - but this table appears to just be the outlook as of the beginning of a game. 

Yes as I stated up front this is your chance to draw a card by a certain point in the game, not the chance to draw a card on any specific draw. You are 100% correct in that if you are looking for one card and have 2 cards left in your deck you have a 50% chance to get it. This is more useful for conversations around deck building (like the Wild Growth example I provided)

Kudos for putting a percentage on how correct I was about 1 out of 2. That's awesome. Sorry for misreading your post.

The percentage is cumulative, not a snapshot. This chart answers a single question:

If I have two of a specific card in my deck, and I will mulligan everything to get it, what is the chance I will have drawn 1 at turn X. Move to your point of interest, and the percentage will tell you the odds of having one by that turn. so when there are 3 cards left in the deck, 99.8% of the time you will have drawn at least one (because its very rare to get both copies in the bottom 3 cards of the deck).

The possibility of drawing the cards you just mulliganned is accounted for as well. You see the possible cards counts from 30 down to 25, and bumps back to 27 because the mulligan cards got added back in.

Very informative graph, I am curious to see the same table for singles (fairly common question because of legendaries and single tech cards)

Saw this topic and thought it would be great to add the percentages when the player is on the coin for everyone's reference.