Hypothetical deck

Hi guys, I'm an old reader of this site and had a theoretical odds question come up in a game I play. I couldn't do the math myself, and it made me think of the Wizard, so I thought I could propose it here and maybe get a concrete answer.

I'm simplifying and adapting the rules of the other game:

Let's say you have a deck of cards, half are black, half are red.

Does the total size of the deck increase the average 'clump' size (i.e. getting multiple red or black cards in a row). In other words, if the deck is 10 cards, the largest clump you can have is 5 black in a row or 5 red in a row. The average sequence probably being something like 2-3 in a row? I'm just using fuzzy math in my head, I'm not a mathematician or statistician, so that might be wrong but it feels correct based on actually playing the game.

Now, if that deck is increased in size to 100 cards, the largest clump you can have is 50 red or 50 black cards in a row. The part I'm not sure about, is whether increasing the maximum size affects the average size in this case. On one hand, it feels logical that if the smallest clump size stays the same, and the largest size increase, it should move that median point a little bit. But, I know math and probability doesn't always jive with what 'feels' logical.

Sure. I'm sure the Wizard could express it more precisely, but here's how I would look at it. Say you have a ten-card deck with an equal proportion of red and black cards. Now, you deal off a "clump' of three black cards. It's now 5 to 2 against the clump growing to a size of 4 with the next card. Then consider a deck of twenty cards, same conditions. This time, it's only 10 to 7 against the clump growing--a much smaller number--so the chances of a given size clump occurring will increase as the deck gets larger. (I hope this explanation made sense :) )

I'm not sure, but I think this effect (however clumsily I've described it) is why you get fewer blackjacks from a shoe than from a single deck.

YES! It did make sense, thank you very much.
I had proposed this question in a discord chat, and someone who knew statistics tried to tell me it would be the same and used some math I didn't fully understand to explain. But, something in the back of my mind kept saying it didn't add up. I had a similar thought as you just did, that as each card comes off the chances change but couldn't express it properly.

Quote: Joeshlabotnik

Sure. I'm sure the Wizard could express it more precisely, but here's how I would look at it. Say you have a ten-card deck with an equal proportion of red and black cards. Now, you deal off a "clump' of three black cards. It's now 5 to 2 against the clump growing to a size of 4 with the next card. Then consider a deck of twenty cards, same conditions. This time, it's only 10 to 7 against the clump growing--a much smaller number--so the chances of a given size clump occurring will increase as the deck gets larger. (I hope this explanation made sense :) )

As an extension of Joeshlabotnik's argument, consider an infinite deck of cards. Then, each time we draw a card from the deck, the ratio of remaining red and black cards remains exactly 1:1. So, our chance of our clump growing always has probability .5. Thus, our average clump size is equal to exactly 2 in the limiting infinite deck case. As we increase our total deck size N, we increase closer and closer to this limiting case where average clump size is 2, as the effect of the current streak (as Joeshalbotnik describes) dissipates.

Yea, in an infinite situation the average clump size becomes 2 as you remove variance over a long enough time line. That's the largest average clump you can get. In the smaller deck the probability is always lower then .5 to continue the streak.
In a 10 card deck, once you draw a single red card, it's 5 to 4 to draw another one, or a probability of .4. So, the average size of a clump is actually bigger in the infinite deck, because any deck smaller then infinite has a probability lower then .5 on each card draw to continue the streak. But, this is a tangent from what I was exploring that is relevant to the hypothetical situation I made, but not as much the original one.

The situation arose from a digital card game that's in beta (called Eternal) that I'm testing, that is very similar to Magic if you are familiar with that. One of the changes they made was to increase minimum deck size to 75 from 60 (in magic).

I proposed that this change, and my own personal experiences, seemed to increase the amount of clumps I was experiencing. Where I'd draw a certain type of card in a row for multiple turns. I just didn't want to go 'feeling' as evidence, as that is unreliable to say the least.

So, the part that we were trying to figure out is if the frequency, and size, of the clumps you encounter had increased going from 60 to 75 cards.

I came here because I'd always loved the wizard's site when I played poker more, and it seemed like a problem much easier then many others I've seen him answer :)