Odds of flopping 3 or 4 Four of a kinds ?
12-19-2014
, 05:20 PM
Given a pair,the probability of flopping quads is C(2,2)*48/C(50,3) = 0.24%. The probability of getting a pair is 1/17. The probability of getting a pair AND then flopping quads is 0.24%/17 = 0.00014.
Now, to answer questions of this type, the question has to be precisely stated. Do we assume the probability that exactly 3 or 3 or more pairs have occurred, or do we include the probability of getting exactly 3 or 3+ pairs, etc. In other words, what is given and what is to be calculated?
I’ll answer the question as follows. In 120 hands, you can expect to be dealt 7 pairs What is the probability that you flop quads at least 3 times given 7 pairs?. The binomial distribution provides the answer as 1.04682E-10 = 0.0000000001 or 1 in 10 billion times.
Could you all have been too drunk to remember exactly what happened.
NOTE; The above focuses on a specific player. For 6 players, the probability that at least one gets 3 or more quads is approximately 6 times the above answer. Still not big enough to not question the liquour holding capacity of all IMO.
Now, to answer questions of this type, the question has to be precisely stated. Do we assume the probability that exactly 3 or 3 or more pairs have occurred, or do we include the probability of getting exactly 3 or 3+ pairs, etc. In other words, what is given and what is to be calculated?
I’ll answer the question as follows. In 120 hands, you can expect to be dealt 7 pairs What is the probability that you flop quads at least 3 times given 7 pairs?. The binomial distribution provides the answer as 1.04682E-10 = 0.0000000001 or 1 in 10 billion times.
Could you all have been too drunk to remember exactly what happened.
NOTE; The above focuses on a specific player. For 6 players, the probability that at least one gets 3 or more quads is approximately 6 times the above answer. Still not big enough to not question the liquour holding capacity of all IMO.
12-19-2014
, 07:53 PM
Quote:
Given a pair,the probability of flopping quads is C(2,2)*48/C(50,3) = 0.24%. The probability of getting a pair is 1/17. The probability of getting a pair AND then flopping quads is 0.24%/17 = 0.00014.
Now, to answer questions of this type, the question has to be precisely stated. Do we assume the probability that exactly 3 or 3 or more pairs have occurred, or do we include the probability of getting exactly 3 or 3+ pairs, etc. In other words, what is given and what is to be calculated?
I’ll answer the question as follows. In 120 hands, you can expect to be dealt 7 pairs What is the probability that you flop quads at least 3 times given 7 pairs?. The binomial distribution provides the answer as 1.04682E-10 = 0.0000000001 or 1 in 10 billion times.
Could you all have been too drunk to remember exactly what happened.
NOTE; The above focuses on a specific player. For 6 players, the probability that at least one gets 3 or more quads is approximately 6 times the above answer. Still not big enough to not question the liquour holding capacity of all IMO.
Now, to answer questions of this type, the question has to be precisely stated. Do we assume the probability that exactly 3 or 3 or more pairs have occurred, or do we include the probability of getting exactly 3 or 3+ pairs, etc. In other words, what is given and what is to be calculated?
I’ll answer the question as follows. In 120 hands, you can expect to be dealt 7 pairs What is the probability that you flop quads at least 3 times given 7 pairs?. The binomial distribution provides the answer as 1.04682E-10 = 0.0000000001 or 1 in 10 billion times.
Could you all have been too drunk to remember exactly what happened.
NOTE; The above focuses on a specific player. For 6 players, the probability that at least one gets 3 or more quads is approximately 6 times the above answer. Still not big enough to not question the liquour holding capacity of all IMO.
I can unequivocally state the following; I myself was not inebriated anywhere close to the point of smearing or marring my recollection of the night. I am a very occasional lightweight, drinking only a few times per year. The others, excluding one, where quite inebriated and I believe this is what leads them to recall that it was actually 4 four of a kinds, not 3 as I have always recalled. I have a very reliable witness who was completely sober on the night in question.
Here are the important details:
We played approximately 120 hands.
In three of those hands I was dealt a pocket pair (KK, JJ, 22). On those particular hands I flopped the 4 of a kind.
I was the only one who got a four of a kind that night.
To answer your second paragraph question, I dealt other pocket pairs that did not quad on the flop. So we'd be calculating probability that 3 or more pairs have occurred, but only 3 of them completed.
That corresponds with the way you calculated it, I believe.
Any other questions or clarifications I am happy to provide.
12-20-2014
, 12:36 AM
Well, what you state is very true.
In fact, I'll often ask people if they think they've ever shuffled a deck of cards so that the order of the 52 shuffled cards matches a shuffle they performed previously sometime in their life. Some people think they have, but most, knowing me and the questions I ask, usually say no. I'll then ask, what if I include all the shuffles of all the people you've ever played cards with? Here I get a 50/50 split, where some will say yes, some will stick on no. Then I'll include all the people they've ever played with, plus all the people those people have ever played with, and at this point, most are willing to wager there has been a duplicate. If they stick on no, I'll ask, if we include every shuffle, since playing cards where invented, and every shuffle that's ever been made, if there has ever been a duplicate and most are now willing to consider the possibility. Of course, the above is very unlikely in that there are something like 80 trillion trillion trillion billion million combinations that the deck can be shuffled in. That makes the chance look infinitesimally small and very unlikely. It does make my 1 in 10 billion look trillions and trillions of times more likely to happen than that, but still very unlikely.
I guess the whole point here is that many events in poker are just that, very unlikely, but given the fact that billions and billions of hands are played, this kind of thing indeed does happen, as you stated, just seeming special when we are involved with them. Though in fact, if you took any two shuffles you've ever played with and asked the odds of having one shuffle followed by the next shuffle, those odds would be huge. Even the odds of any shuffle are 80 trillion trillion... to 1. Of course, the shuffle must end somewhere.
I knew the odds of my flopped 4 of a kinds was large, but had to idea it'd be out of the hundreds of millions to 1. I always told people that I was probably more likely to win the lottery, but never could give a real number.
Thanks for providing the math. I am looking to take a math course in the new year to try and start understanding probability, or at least how to begin calculating them.
...
A few related points to how the cards unfolded. On the very first hand played that night, I was dealt JJ. A friend of mine in early position made it 3x to go. I thought it was too small a raise in this game to be anything very good so I 3 bet him, to which he shoved. I guess a fold could be in order here, but this was a few years before I knew such things.
My brother dealt the flop. J in the window, and one more when he fanned it out.
Later, another player joined the game, and of the lot of us I'd venture to say he was the best player at the table. I was dealt deuces against him. Flopped quads and went to showdown. He was amazed and talking about the odds of flopping the hand. Over the drunkenness permeating the room, it was conveyed that it was the second time it had happened.
Flash forward about four hours later, and we were playing the last hand before the end of re-buys; it was getting late. The board paired Kings, and to my amazement, the four players in front of me were all going all in, probably figuring if they lost they could re-buy, and they were mostly under a full stack anyway. After a few minutes of them making side pots, I whispered to my brother "Guess what I got?' and it was half heard by the room, at which point I flipped them over, pocket Kings, and pandemonium erupted.
Thanks again for taking the time to consider the answer. Appreciate it.
In fact, I'll often ask people if they think they've ever shuffled a deck of cards so that the order of the 52 shuffled cards matches a shuffle they performed previously sometime in their life. Some people think they have, but most, knowing me and the questions I ask, usually say no. I'll then ask, what if I include all the shuffles of all the people you've ever played cards with? Here I get a 50/50 split, where some will say yes, some will stick on no. Then I'll include all the people they've ever played with, plus all the people those people have ever played with, and at this point, most are willing to wager there has been a duplicate. If they stick on no, I'll ask, if we include every shuffle, since playing cards where invented, and every shuffle that's ever been made, if there has ever been a duplicate and most are now willing to consider the possibility. Of course, the above is very unlikely in that there are something like 80 trillion trillion trillion billion million combinations that the deck can be shuffled in. That makes the chance look infinitesimally small and very unlikely. It does make my 1 in 10 billion look trillions and trillions of times more likely to happen than that, but still very unlikely.
I guess the whole point here is that many events in poker are just that, very unlikely, but given the fact that billions and billions of hands are played, this kind of thing indeed does happen, as you stated, just seeming special when we are involved with them. Though in fact, if you took any two shuffles you've ever played with and asked the odds of having one shuffle followed by the next shuffle, those odds would be huge. Even the odds of any shuffle are 80 trillion trillion... to 1. Of course, the shuffle must end somewhere.
I knew the odds of my flopped 4 of a kinds was large, but had to idea it'd be out of the hundreds of millions to 1. I always told people that I was probably more likely to win the lottery, but never could give a real number.
Thanks for providing the math. I am looking to take a math course in the new year to try and start understanding probability, or at least how to begin calculating them.
...
A few related points to how the cards unfolded. On the very first hand played that night, I was dealt JJ. A friend of mine in early position made it 3x to go. I thought it was too small a raise in this game to be anything very good so I 3 bet him, to which he shoved. I guess a fold could be in order here, but this was a few years before I knew such things.
My brother dealt the flop. J in the window, and one more when he fanned it out.
Later, another player joined the game, and of the lot of us I'd venture to say he was the best player at the table. I was dealt deuces against him. Flopped quads and went to showdown. He was amazed and talking about the odds of flopping the hand. Over the drunkenness permeating the room, it was conveyed that it was the second time it had happened.
Flash forward about four hours later, and we were playing the last hand before the end of re-buys; it was getting late. The board paired Kings, and to my amazement, the four players in front of me were all going all in, probably figuring if they lost they could re-buy, and they were mostly under a full stack anyway. After a few minutes of them making side pots, I whispered to my brother "Guess what I got?' and it was half heard by the room, at which point I flipped them over, pocket Kings, and pandemonium erupted.
Thanks again for taking the time to consider the answer. Appreciate it.