primrose
Visionary
Bronze Level
Last night when going to bed, I asked myself this practically-irrelevant-but-interesting-to-me question: how likely is it that there is a board (i.e., the five cards in the middle, in their exact order) that has never been seen before, throughout all games throughout human history?
So let's answer it using MATH. First, the number of boards when considering order is simply 52*51*50*49*48 = 311 875 200, which is roughly 300 million. Now we need to estimate the number of boards that have ever been revealed. I decided that online games don't count for this because well what if a program like Equilab actually "reveal" a gazillion boards to compute a probability; surely those don't count, but if we say that online poker rooms do count, that seems like a slippery slope. So physical cards only.
I asked GPT-4 to estimate the number of physical boards ever revealed then tweaked its numbers. It had some nonsensical assumptions, like that average active player sees 50 flops per year (wtf?). I think 10000 is a better estimate here, although then we can divide it by 10 because most flops are shown to around 9 players, and by 10 again because only about a tenth of games go to the River. (We're going to be off by many OOMs here anyway so I don't think we need more precise numbers for this point.) So, assuming there's around 4 million active live poker players alive today, we get around 100 * 4 000 000 Rivers per year. Assuming they've been playing for 50 years, we have 20 000 000 000, or 20 billion, boards seen. Now of course, poker has been around a lot longer than 50 years, but actually in such cases usually the modern times dominate the calculation because of population growth. Roughly a tenth of all people who have ever lived all throughout history are alive today! So I think just making that 30 000 000 000 to account for all prior history is a reasonable guess.
Alright now with 30 billion boards revealed vs. 300 million possible boards, we know that most boards have been revealed many times -- the average one around 100 times, in fact. Which means that next time you play live and get to the River, you can feel a special spiritual connection with a hundred other groups of players throughout history or something. But that doesn't mean that every board has been revealed! If you randomize 200 numbers between 1 and 100, it's actually not at all likely that you'll have gotten each number at least once even though you'll have gotten most of them twice. Ofc since mathematicians love to study elegant sounding things, this is a known problem called the Coupon's Collector Problem, and we know that the expected number of trials to get each and every one of N numbers at least once (given repeated uniform picks) grows asymptotically with N ln(N). Now ln(300 000 000) = 19.519..., so around 20. Which means we'd expect that around 20*300 000 000 = 6 000 000 000 boards have to be revealed until each one is revealed at least once. Since the real number is around 5 times that (30 vs. 6 billion), that means that, no, you cannot see a new board; it's highly likely that every possible board has been revealed already.
... well, except that the number here is only larger by a factor of 5 and I'd expect my estimates to be off by a lot more than that, so the real answer is that idk. Although my gut says we've probably underestimated the number of boards, so I'm still leaning yes.
At least I know one thing for sure, which is that you will never get the 5 minutes of your life back that you just wasted reading about this problem.
So let's answer it using MATH. First, the number of boards when considering order is simply 52*51*50*49*48 = 311 875 200, which is roughly 300 million. Now we need to estimate the number of boards that have ever been revealed. I decided that online games don't count for this because well what if a program like Equilab actually "reveal" a gazillion boards to compute a probability; surely those don't count, but if we say that online poker rooms do count, that seems like a slippery slope. So physical cards only.
I asked GPT-4 to estimate the number of physical boards ever revealed then tweaked its numbers. It had some nonsensical assumptions, like that average active player sees 50 flops per year (wtf?). I think 10000 is a better estimate here, although then we can divide it by 10 because most flops are shown to around 9 players, and by 10 again because only about a tenth of games go to the River. (We're going to be off by many OOMs here anyway so I don't think we need more precise numbers for this point.) So, assuming there's around 4 million active live poker players alive today, we get around 100 * 4 000 000 Rivers per year. Assuming they've been playing for 50 years, we have 20 000 000 000, or 20 billion, boards seen. Now of course, poker has been around a lot longer than 50 years, but actually in such cases usually the modern times dominate the calculation because of population growth. Roughly a tenth of all people who have ever lived all throughout history are alive today! So I think just making that 30 000 000 000 to account for all prior history is a reasonable guess.
Alright now with 30 billion boards revealed vs. 300 million possible boards, we know that most boards have been revealed many times -- the average one around 100 times, in fact. Which means that next time you play live and get to the River, you can feel a special spiritual connection with a hundred other groups of players throughout history or something. But that doesn't mean that every board has been revealed! If you randomize 200 numbers between 1 and 100, it's actually not at all likely that you'll have gotten each number at least once even though you'll have gotten most of them twice. Ofc since mathematicians love to study elegant sounding things, this is a known problem called the Coupon's Collector Problem, and we know that the expected number of trials to get each and every one of N numbers at least once (given repeated uniform picks) grows asymptotically with N ln(N). Now ln(300 000 000) = 19.519..., so around 20. Which means we'd expect that around 20*300 000 000 = 6 000 000 000 boards have to be revealed until each one is revealed at least once. Since the real number is around 5 times that (30 vs. 6 billion), that means that, no, you cannot see a new board; it's highly likely that every possible board has been revealed already.
... well, except that the number here is only larger by a factor of 5 and I'd expect my estimates to be off by a lot more than that, so the real answer is that idk. Although my gut says we've probably underestimated the number of boards, so I'm still leaning yes.
At least I know one thing for sure, which is that you will never get the 5 minutes of your life back that you just wasted reading about this problem.
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