The Bridge Problem

I know you guys like math and are a lot better at it than me so I thought of a neat puzzle. let me preface by saying I have no idea of the answer.

The problem goes as follows:

you start with an ordinary deck of 52 cards on round 1
for each round, deal piles of 13 cards and discard the extras (for the first round there will of course always be 4 even piles)
look at each hand individually and take out the best bridge suit then discard the rest. (Example: As 10s 3s 4d 8d 9d 3d Jh Qh 2h 7h 6c 7c, set aside the hearts since there are 4 diamonds and hearts but the highest heart (queen) beats the highest diamond (9) the discard the clubs, diamonds, and spades)
after all hands for that round have been resolved, take the discards and shuffle them (or don't I'm not your dad) then start a new round and deal piles of 13
remember to keep these hands separate from the last round and resolve the hands for this round
repeat until there are 12 or fewer discards
the round which has the single highest hand wins. In the event of a tie, buy a lottery ticket

The question is which round wins most often?

My hypothesis would be the last round since it has all of the suits that haven't been picked yet. After 5 games they were all 4 round games but can range anywhere from 1-5. After those games my results were rounds 2-1-1-2-3 being the victors. I'm now led to believe that round 1 will win most of the time because it has the most amount of options, but a game where it is lopsided after round 1 usually results in higher hands afterward.

I'm interested to see what you guys come up with and your methodology. I hypothesize a way where you just calculate the probability of getting 4/13 or a 6/13 hand etc. combined with the probability of it being a certain suit and that would avoid computer simulations. I'm kinda afraid of excel/programming tho so gl figuring this out!

How many games would you have to play to become a millionaire from lottery winnings?

It looks like there are too many possibilities to work it out mathematically, as opposed to through simulation.

I did figure out that there will be between 4 and 10 deals - you can get 4 if, somehow, you deal a hand where all 13 cards are the same suit in each round; since every hand will have at least 4 cards of one suit (otherwise the most you can have in a hand is 4 suits with 3 cards each, but that's only 12 cards), 4 cards will be discarded in each deal; after 10 deals, there are at most 12 cards left.

What I got in simulation was this:
The most likely result is the last round winning after 9 deals; this happened 8.5% of the time.
The other 8 possible results in a 9-deal game each happened about 5.75% of the time.
Each of the 8 results in an 8-deal game happened about 5.25% of the time (the last round wins slightly more often than the others).
The last round winning in a 10-deal game happened about 0.7% of the time; no other result happened more than 0.2%.
I noticed that, except for the last round, each of the rounds for any particular number of rounds were equally likely to win; the gap between the last round and the others increased as the number of rounds did - 30% of the wins in a 10-round game happened in the 10th round, and each of the other 9 rounds won around 8% of the time.

Quote: ThatDonGuy

It looks like there are too many possibilities to work it out mathematically, as opposed to through simulation.

I did figure out that there will be between 4 and 10 deals - you can get 4 if, somehow, you deal a hand where all 13 cards are the same suit in each round; since every hand will have at least 4 cards of one suit (otherwise the most you can have in a hand is 4 suits with 3 cards each, but that's only 12 cards), 4 cards will be discarded in each deal; after 10 deals, there are at most 12 cards left.

What I got in simulation was this:
The most likely result is the last round winning after 9 deals; this happened 8.5% of the time.
The other 8 possible results in a 9-deal game each happened about 5.75% of the time.
Each of the 8 results in an 8-deal game happened about 5.25% of the time (the last round wins slightly more often than the others).
The last round winning in a 10-deal game happened about 0.7% of the time; no other result happened more than 0.2%.
I noticed that, except for the last round, each of the rounds for any particular number of rounds were equally likely to win; the gap between the last round and the others increased as the number of rounds did - 30% of the wins in a 10-round game happened in the 10th round, and each of the other 9 rounds won around 8% of the time.
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Wow that’s super interesting. I thought for sure the correlation between round and win% would be a u curve or upward trend, or at the very least a line. Since if you deal a perfect bridge hand, the other rounds won’t exist therefore the first round would be slightly higher. It makes since for a 5 round game that hands 1-9 would have the same odds seeing that by design almost all hands will be four cards. But a typical 8-9 (4-2-1-1 or 4-2-2-1) hand game has so much more variability. I appreciate your results a lot and it put a smile on my face when I read them. I’m sure this means round 4 wins most often since I’m guessing a 5 round game happens so little that it wouldn’t make the podium.

Questions time: Do you have screenshots of results? You don’t have to redo the sim if it takes hours but I’m curious.
Did your sim shuffle after every hand or after every round (of 4 hands on round 1) I wonder if results would differ if you didn’t shuffle. I kinda doubt tho

Quote: richodude

Wow that’s super interesting. I thought for sure the correlation between round and win% would be a u curve or upward trend, or at the very least a line. Since if you deal a perfect bridge hand, the other rounds won’t exist therefore the first round would be slightly higher. It makes since for a 5 round game that hands 1-9 would have the same odds seeing that by design almost all hands will be four cards. But a typical 8-9 (4-2-1-1 or 4-2-2-1) hand game has so much more variability. I appreciate your results a lot and it put a smile on my face when I read them. I’m sure this means round 4 wins most often since I’m guessing a 5 round game happens so little that it wouldn’t make the podium.

Questions time: Do you have screenshots of results? You don’t have to redo the sim if it takes hours but I’m curious.
Did your sim shuffle after every hand or after every round (of 4 hands on round 1) I wonder if results would differ if you didn’t shuffle. I kinda doubt tho
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First of all, even if I did deal a hand where every card is the same suit in the first round, I would continue until there were fewer than 12 cards left; I wouldn't stop at 1 round.

No, I don't have screenshots of results; besides, I pretty just told you what the displayed results were.
I am a little confused about your shuffling questions. You only shuffle before you deal a round of hands. There is no reason to shuffle after each hand; you need to see all of the hands before you know which hand is the best and, consequently, which cards to remove.

Quote: ThatDonGuy

First of all, even if I did deal a hand where every card is the same suit in the first round, I would continue until there were fewer than 12 cards left; I wouldn't stop at 1 round.

No, I don't have screenshots of results; besides, I pretty just told you what the displayed results were.
I am a little confused about your shuffling questions. You only shuffle before you deal a round of hands. There is no reason to shuffle after each hand; you need to see all of the hands before you know which hand is the best and, consequently, which cards to remove.
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To clear up my definitions
Hand - A group of 13 cards
Round - The hands that make up the cards remaining in the deck. Ex. Round 2 usually has 2 hands, and round 1 always has 4 hands.
Game - All the rounds played until the deck has less than 13 cards

My question was say on round one your hands were 7clubs, 5dias, 4spaded, 6hearts. This means you have 30 discards to start round 2. Would it make a difference if those 30 discards were shuffled before the round starts or if you kept dealing in the order you discarded them

Quote: richodude

Quote: ThatDonGuy

First of all, even if I did deal a hand where every card is the same suit in the first round, I would continue until there were fewer than 12 cards left; I wouldn't stop at 1 round.

No, I don't have screenshots of results; besides, I pretty just told you what the displayed results were.
I am a little confused about your shuffling questions. You only shuffle before you deal a round of hands. There is no reason to shuffle after each hand; you need to see all of the hands before you know which hand is the best and, consequently, which cards to remove.
link to original post

To clear up my definitions
Hand - A group of 13 cards
Round - The hands that make up the cards remaining in the deck. Ex. Round 2 usually has 2 hands, and round 1 always has 4 hands.
Game - All the rounds played until the deck has less than 13 cards

My question was say on round one your hands were 7clubs, 5dias, 4spaded, 6hearts. This means you have 30 discards to start round 2. Would it make a difference if those 30 discards were shuffled before the round starts or if you kept dealing in the order you discarded them
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OOPS - I played it wrong.
In my version, if your four hands were 7 clubs, 5 diamonds, 4 spades, and 6 hearts, you only removed the 7 clubs in the first hand, and shuffled the other 45 cards.

In the correct version, there cannot be more than 5 rounds:
Round 1: 4 hands of 4, leaving 36
Round 2: 2 hands of 4, leaving 28
Round 3: 2 hands of 4, leaving 20
Round 4: 1 hand of 4, leaving 16
Round 5: 1 hand of 4, leaving 12

I will see if I can change the simulation to do it your originally stated way.

And here is the updated version, in handy table format
The first column is how many rounds were played, the second is the round with the best hand, the third is what percentage of the games that had that many rounds had that round number as the best play, and the fourth is what percentage overall were that many rounds with that particular best round (e.g. 26.473% of the time, there were 4 rounds played, and the 4th round had the best hand)
This is based on a simulation of 34,033,670 games

Rounds	Best Round	% in that many rounds	% overall
2	1	6.904	0.002
2	2	93.096	0.022
3	1	30.419	2.998
3	2	48.224	4.752
3	3	21.357	2.105
4	1	17.742	12.187
4	2	28.387	19.499
4	3	15.332	10.532
4	4	38.539	26.473
5	1	21.763	4.646
5	2	29.990	6.402
5	3	11.343	2.421
5	4	17.361	3.706
5	5	19.543	4.172
6	1	45.606	0.039
6	2	40.870	0.035
6	6	13.524	0.012

The numbers look like they are all over the place

Those are even more interesting than the 1st time! Seems like the 2nd round wins most often followed by the last. All very different depending on the number of rounds! Didn’t realize that is round 1 has 3 hands of 5 and a hand of 4 leaving 33, round 2 having 2 hands of 4 leaving 25, round 3 having a hand of 4 leaving 21, round 4 having a hand of 4 leaving 17, round 5 having a hand of 4 leaving 13, gives you a 6th round. Rounds 1 and/or 2 have to have a 5 card hand along with round 6 having whatever. That means rounds 3-5 never win on a 6 round game.