RACK-O: Why everyone seems to finish one card shy of winning.

• Most games of RACK-O are destined to end up neck-and-neck (i.e. all 4 players have just 1 card they are looking for).

RACK-O is a card game with a simple objective: arrange your 10-card hand in numerical order from lowest to highest. I'll leave the finer details of the rules for you to read on your own. But a sample hand might be played like:

• Each player is a dealt a 10 card hand in random order from a deck with cards 1 through 60. For instance, you might start with the hand: [2, 10, 50, 51, 52, 53, 9, 26, 29, 31]

• You next have the option to replace a single card in your hand from one of the two decks in the center of the table, but you cannot move cards around in your hand. 

• You can choose the top card from the face-down pile or the top-card of the face-up pile. Let's say the face-up pile shows 12. Let's say you choose that card, and choose to replace the 50 card in your hand. Your hand then becomes slightly closer to being in ascending order:  [2, 10, 12, 50, 51, 52, 53, 9, 26, 29, 31].

•  You go through rounds of drawing and replacing cards in your hand in order, until your hand ends up in an ascending sequence. E.g.  [2, 10, 12, 15, 16, 17, 20, 26, 29, 31] The first person to do so wins the round and says RACKO.

What usually happens at game-end is this:

• When someone calls RACKO, everyone else playing was only 1 or 2 cards away from their own RACKO.

This seems counterintuitive. How can it be that when one person wins, it so happens that everyone else was also very close to winning?

The answer is underwhelming:

• It is very easy to get to 1 or 2 cards away from a RACKO. And very hard to get that final card.

So if we could see everyone's hands during a RACKO game (which typically ends at about 8 rounds of play), what we would (on average) see is:

• After 5-6 rounds, everyone is  1 or 2 cards away from a RACKO. The remaining rounds are more or less spent with players waiting to see the one card that they need appear on the face-up pile; or they get lucky and draw said card from the face-down pile.*

*Note: this ignores any strategy. Typically players will try to replace the cards at the beginning of their sequence with the lowest they can find, and the highest they can find at the end. Any strategy in RACKO, due to the small number of rounds required to win, has a small effect at the margin.

Let's assume we have an average amount of luck so that we start the game with 4 cards in order. What next?

Here we do some back-of-the-envelope maths. We aren't after anything precise here, we are after an intuitive understanding of why RACKO games seem to all end the same.

We have 4 cards in order. That means we have 6 cards we need to replace. On average, those six cards are going to evenly span the range from 0 to 60. This is really important: it means that in the first round, we are guaranteed to draw a card that we need.

Still don't see why? Consider an example hand:

[X, 21, X, 34, X, 50, X, 55, X, 60]

the above hand is equivalent to: Get me any card that is not 21, 34, 50, 55, or 60. I.e. 55/60 or 92% of the cards in the deck. So we have a 92% chance of drawing a card we need in the first round.

It is actually considerably better than that though, remember that there is both a face-down and a face-up pile and we can choose a card from either. That gives us two chances, so the chance of getting a card we need is actually 1 - (1-0.92)^2 = 0.99. So 99%!

So let's keep this tally: we now have 5 cards we need, and one round has passed.

Again lets use our intuition. We had 6 gaps that spanned the range from 0 to 60. E.g., we needed one card in either [1 to 10], [11 to 20], [21 to 30], [31 to 40], [41 to 50], [51 to 60]. 

Unless we got very unlucky (1% chance), we just filled one of those gaps in the first round. So now we need a card that is from 50 out of the 60 cards in the deck (5 of the 6 gaps listed above). Again we have two chances at a draw, so our chance of drawing a card we need is:

• 1 - (1-50/60)^2 = 97%

So after 2 rounds, we have 6 cards in order and 4 out of order.

In subsequent draws, our chance of drawing a card we need is 1 - (1-40/60)^2 = 88%. Assuming we draw that card, the next round then has a 1 - (1-30/60)^2 = 75% chance of getting the card we need. And to get to the point where we only have 1 card left to replace? That has a 55% chance of happening.

So on average, it takes 2 rounds to get to the point where we have only 4 cards out of order.

Then after that, we expect roughly 4 more rounds to get to the final place in RACKO where we have just 1 card left:

1/0.88 + 1/0.75 + 1/0.55 = 4.28 

So 4ish rounds to get from 4 cards left to 1 card left.

(Note 1/p is the expected number of rounds to draw your missing card: https://en.wikipedia.org/wiki/Geometric_distribution)

We had 2 rounds to get from the average starting hand to 4 cards left; and 4 rounds to get from 4 left to 1.

So on average, it takes 6 rounds to get from the average starting hand to the point where we are 1 card away from RACKO.

The average player will take 6 rounds to get to the point where they are one card away from a RACKO. What about that final card? Well the chance of that is going to depend on the specific hand, and the gap you need to fill (if the final card you need has to fall between 10 and 30, you are in much better shape than if the gap is between 10 and 15). But if we start with our "average" hand where the gaps were all even intervals of 10, then this probability is 1/6. Meaning it takes on average 6 rounds for you to draw that final card.

But remember we have 2-4 players. So really the chance that any of the 4 players finishes RACKO is much higher (https://en.wikipedia.org/wiki/Bernoulli_trial): about 40%   

(40% is just the binomial distribution result with k=1 successes, 4 trials, and p=1/6).

So it should take 2-3 rounds (once we reach end-game with 1 card left) in a 4 person game to have someone call RACKO. Hence why RACKO feels so neck and neck all the time:

The average player gets to having 1 card left in just 6 rounds; thereafter, 2-3 rounds goes by where all players are digging for their final card.

This is why most RACKO games take 8-9 rounds to finish, and the last third of the game is effectively guaranteed to be spent neck-and-neck.

*for a short-handed (i.e. 2 person) game, the number of rounds will be longer. But in fact the rules of RACKO has you reduce the deck to just 40 cards to actually counteract this effect so the game is not prolonged.