Quizzes & Puzzles25 mins ago
QI
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The QI repeat broadcast tonight.
Re the deck of cards shuffling; am I the only one who thought that SF's claim was nonsense?
Re the deck of cards shuffling; am I the only one who thought that SF's claim was nonsense?
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No you are not the only one. It is incorrect.
Like others, I knew the answer was 52 factorial before it was stated and was very surprised (and annoyed) they asserted it was certain every card set as the result of shuffling was unique and had definitely never occurred before.
Reading this thread, there are opposing views which seem not to hinge on how many combinations there are – i.e. 52! but rather practical human sense probability versus mathematical probability.
Mathematically, even if you add a trillion 0s to the odds – e.g. 52!^10^12 to 1, and only randomly arrange a pack of cards twice in the history of mankind, it is still not certain they will not be the same. There is no “almost impossible” mathematically – it either cannot (impossible), can (possible) or will occur (certain). This example is in the middle (possible) category – it doesn’t matter how unlikely it is unless the probability is 0 (impossible).
Practically speaking, billions of people have shuffled cards since standard cards were invented – some of them do it all the time for a job. I don’t think it is unfair to estimate there have been 10^15 – 10^17 shuffles all time. You don’t have to exhaust all combinations before you get an identical shuffled card set. Just 2 tries halves the odds. And of course every time you shuffle the cards, there is a new result which can be added to the card sets which have occurred which may be duplicated Every subsequent card set result may match that also. Fair enough 8*10^67 is a big number, but quadrillions of card sets is pretty big too. That doesn’t mean I’m saying a card set match has occurred, but to say it definitely hasn’t is wrong.
And of course there are factors that skew the probabilities in the real world:
- Newly opened packs are ordered the same
- Many individuals shuffle in the same way each time (type of shuffle, repetitions etc). That doesn’t mean identical but in the case of an ordered pack this dramatically alters the odds
To sum up:
- Mathematically the assertion that no resultant shuffled decks are the same is definitely and completely wrong.
- Practically speaking is may be unlikely, but it isn’t impossible.
This doesn’t mean I think I can rent an expo centre and prove it with a desk of cards to squarebear in front of everyone – if anyone thinks that’s what I’m saying they haven’t understood it (perhaps my fault for poor explanation). If you take anything away from the above ignore the numbers – it’s the concept of mathematical v practical probability.
Like others, I knew the answer was 52 factorial before it was stated and was very surprised (and annoyed) they asserted it was certain every card set as the result of shuffling was unique and had definitely never occurred before.
Reading this thread, there are opposing views which seem not to hinge on how many combinations there are – i.e. 52! but rather practical human sense probability versus mathematical probability.
Mathematically, even if you add a trillion 0s to the odds – e.g. 52!^10^12 to 1, and only randomly arrange a pack of cards twice in the history of mankind, it is still not certain they will not be the same. There is no “almost impossible” mathematically – it either cannot (impossible), can (possible) or will occur (certain). This example is in the middle (possible) category – it doesn’t matter how unlikely it is unless the probability is 0 (impossible).
Practically speaking, billions of people have shuffled cards since standard cards were invented – some of them do it all the time for a job. I don’t think it is unfair to estimate there have been 10^15 – 10^17 shuffles all time. You don’t have to exhaust all combinations before you get an identical shuffled card set. Just 2 tries halves the odds. And of course every time you shuffle the cards, there is a new result which can be added to the card sets which have occurred which may be duplicated Every subsequent card set result may match that also. Fair enough 8*10^67 is a big number, but quadrillions of card sets is pretty big too. That doesn’t mean I’m saying a card set match has occurred, but to say it definitely hasn’t is wrong.
And of course there are factors that skew the probabilities in the real world:
- Newly opened packs are ordered the same
- Many individuals shuffle in the same way each time (type of shuffle, repetitions etc). That doesn’t mean identical but in the case of an ordered pack this dramatically alters the odds
To sum up:
- Mathematically the assertion that no resultant shuffled decks are the same is definitely and completely wrong.
- Practically speaking is may be unlikely, but it isn’t impossible.
This doesn’t mean I think I can rent an expo centre and prove it with a desk of cards to squarebear in front of everyone – if anyone thinks that’s what I’m saying they haven’t understood it (perhaps my fault for poor explanation). If you take anything away from the above ignore the numbers – it’s the concept of mathematical v practical probability.
With respect to both squarebear and Khandro, I do not believe you have a full grasp of the mathematical concepts involved here. (I am not saying that you both have identical positions as each other on this by the way).
1) (for Khandro) If the odds are 6 to 1 of getting a 6 when rolling a dice, and I permit you 1 roll, you have a 1 in 6 chance. If I allow you 2 rolls, you have a 1 in 3 chance.
It's true each time you roll the dice it is 1 in 6 for that specific roll (the probability term is independent event). But you wouldn't say if I gave you 100 rolls you only had a 1 in 6 chance of getting a 6 a least once.
2) And in the cards example, these are not independent events. Each shuffled deck result affects the likelihood of subsequent shuffle matches. It is not that we have 10 goes at guessing a number between 1 and 1000 (100 to 1). We have 10 goes at guessing 10 numbers between 1 and 1000 (10 to 1).
This is a key point, the odds for the card match shrinks in 2 different dimensions at the same time drastically reducing the odds. So it's not a 10^17 in 8*10^67 chance, it's 10^17 in 8*10^67/10^17. This is a massive difference. Still very very unlikely, but not impossible which is the key point and why people are complaining (on here and other places also). Especially given the pedantic nature of a programme like QI.
The above pertains to the practical probability side. On the mathematical theory side, just 1 chance in 10^100 (a googol) is possible. Or 1 in 10 followed by atoms in the universe number of 0s. Impossible means no chance in whatever number of possibilities. Anyone that is unable to recognise this doesn't understand what impossible means.
I suspect everyone is unlikely to agree, and sometimes the Internet isn't the best place to hammer these things out as people may be more interested in defending the views they hold and less concerned with getting the right answer. Personally, I am only interested in specific constructive responses which demonstrate a grasp of the mathematical concepts.
1) (for Khandro) If the odds are 6 to 1 of getting a 6 when rolling a dice, and I permit you 1 roll, you have a 1 in 6 chance. If I allow you 2 rolls, you have a 1 in 3 chance.
It's true each time you roll the dice it is 1 in 6 for that specific roll (the probability term is independent event). But you wouldn't say if I gave you 100 rolls you only had a 1 in 6 chance of getting a 6 a least once.
2) And in the cards example, these are not independent events. Each shuffled deck result affects the likelihood of subsequent shuffle matches. It is not that we have 10 goes at guessing a number between 1 and 1000 (100 to 1). We have 10 goes at guessing 10 numbers between 1 and 1000 (10 to 1).
This is a key point, the odds for the card match shrinks in 2 different dimensions at the same time drastically reducing the odds. So it's not a 10^17 in 8*10^67 chance, it's 10^17 in 8*10^67/10^17. This is a massive difference. Still very very unlikely, but not impossible which is the key point and why people are complaining (on here and other places also). Especially given the pedantic nature of a programme like QI.
The above pertains to the practical probability side. On the mathematical theory side, just 1 chance in 10^100 (a googol) is possible. Or 1 in 10 followed by atoms in the universe number of 0s. Impossible means no chance in whatever number of possibilities. Anyone that is unable to recognise this doesn't understand what impossible means.
I suspect everyone is unlikely to agree, and sometimes the Internet isn't the best place to hammer these things out as people may be more interested in defending the views they hold and less concerned with getting the right answer. Personally, I am only interested in specific constructive responses which demonstrate a grasp of the mathematical concepts.
Square Bear, I have e-mailed Professor David Spiegelhalter at the Nat.Stat.Lab.Cambridge & he very kindly sent me an ans.
He said you're an idiot.
No,no,no, only joking.As you can see on other thread, I'm having trouble' pasting' reply here..(Whose the idiot now.)
If you'll allow me to precis;
"Of course,it's not impossible,just very unlikely"
Then a load of Math, which included your figures.
Ending"you could shuffle cards every second from now to the end of the universe without attaining match,EQUALLY it could happen to-morrow.
I'll work on e-mail now.
He said you're an idiot.
No,no,no, only joking.As you can see on other thread, I'm having trouble' pasting' reply here..(Whose the idiot now.)
If you'll allow me to precis;
"Of course,it's not impossible,just very unlikely"
Then a load of Math, which included your figures.
Ending"you could shuffle cards every second from now to the end of the universe without attaining match,EQUALLY it could happen to-morrow.
I'll work on e-mail now.
There is generally considered to be more than one type of impossibility though, which I think is the issue here.
A statistical impossibility (which is what the cards are) is when something is so unlikely to happen that it can be generally considered to be impossible. For example, if you were to throw 10 million dice the chances of all of them landing on a 6 is so tiny it would generally be considered impossible, but there is of course a chance it could happen regardless.
Or an absolute impossibility, the above example with the dice isn't an absolute impossibility because there is a chance it could happen, just like the deck of cards repeating.
Both the above are perfectly valid impossibles in stats IMO.
A statistical impossibility (which is what the cards are) is when something is so unlikely to happen that it can be generally considered to be impossible. For example, if you were to throw 10 million dice the chances of all of them landing on a 6 is so tiny it would generally be considered impossible, but there is of course a chance it could happen regardless.
Or an absolute impossibility, the above example with the dice isn't an absolute impossibility because there is a chance it could happen, just like the deck of cards repeating.
Both the above are perfectly valid impossibles in stats IMO.
I will ask again ....define a shuffle.
If you use a newly opened pack give a pack of cards to 8 billion people they cut the cards to start a shuffle. Dived that by 52 on average......
153846153.846 people will have started their shuffle with the same card.
do another cut and
2958579.88165 people will have cut at the same card
and again
56895.7669548 people will cut the same card
and again
1094.14936452 people will cut the same card
one more
21.0413339331 people.
Does 5 cuts count as a shuffle?
Also notice that they start with a pack of cards that has already been shuffled
I think that was deliberate.
If you use a newly opened pack give a pack of cards to 8 billion people they cut the cards to start a shuffle. Dived that by 52 on average......
153846153.846 people will have started their shuffle with the same card.
do another cut and
2958579.88165 people will have cut at the same card
and again
56895.7669548 people will cut the same card
and again
1094.14936452 people will cut the same card
one more
21.0413339331 people.
Does 5 cuts count as a shuffle?
Also notice that they start with a pack of cards that has already been shuffled
I think that was deliberate.
I thought it was answered that 7 shuffles enables any possibility to form?
Sorry but I still think that it is impossible for anything as crazy as throwing more dice than grains of sand in billions of Sahara Deserts and expecting them to all land on a 6 is perfectly possible. No wonder the casinos I saw in Las Vegas were all dripping in gold and Tiffany glass with such optimistic people.
But I guess different people believe different things.
Sorry but I still think that it is impossible for anything as crazy as throwing more dice than grains of sand in billions of Sahara Deserts and expecting them to all land on a 6 is perfectly possible. No wonder the casinos I saw in Las Vegas were all dripping in gold and Tiffany glass with such optimistic people.
But I guess different people believe different things.
ChuckFickens makes a valid point, though the problem with statistical impossibility is that the threshold for when something qualifies as arbitrary. 10^-50 is sometimes used but there is no defined probability.
Also, the 10 million dice example is very different to the cards scenario. Each time the dice as ejected the results are random and have no impact on the likelihood of subsequent throws. The cards situation is fundamentally different and has a "memory" of previous attempts which the dice don't. Over time the probability of a match becomes higher and higher until it gets to certain.
I also think the odds are better than the 10^-50 I've quoted and wouldn't qualify (though as I say it has an arbitrary nature). This to a large degree depends on how many shuffles have occurred. I assume we are defining each shuffle as very thorough and the starting state of each desk random (which doesn't happen in the real world).
Unlikely perhaps. But QI has a history of being the most excessively exact and pedantic programme (though I like it), and the way in which this problem was present was inaccurate in my opinion.
Also, the 10 million dice example is very different to the cards scenario. Each time the dice as ejected the results are random and have no impact on the likelihood of subsequent throws. The cards situation is fundamentally different and has a "memory" of previous attempts which the dice don't. Over time the probability of a match becomes higher and higher until it gets to certain.
I also think the odds are better than the 10^-50 I've quoted and wouldn't qualify (though as I say it has an arbitrary nature). This to a large degree depends on how many shuffles have occurred. I assume we are defining each shuffle as very thorough and the starting state of each desk random (which doesn't happen in the real world).
Unlikely perhaps. But QI has a history of being the most excessively exact and pedantic programme (though I like it), and the way in which this problem was present was inaccurate in my opinion.
I need to post a correction here:
"This is a key point, the odds for the card match shrinks in 2 different dimensions at the same time drastically reducing the odds. So it's not a 10^17 in 8*10^67 chance, it's 10^17 in 8*10^67/10^17. This is a massive difference."
It's true the odds scale down in 2 directions as the shuffles continue, massively reducing them and something I don't think all who watched QI grasped, but 10^17 in 8*10^67/10^17 isn't correct because it gives the impression that for each of the 10^17 shuffles there were 10^17 matches. Whilst that is correct for the 10^17th shuffle, it wasn't for the 1st, 2nd etc where there were only 1,2 etc matches to choose from. Recursion is needed for the proper equation.
Apologies for that (and the typos!).
"This is a key point, the odds for the card match shrinks in 2 different dimensions at the same time drastically reducing the odds. So it's not a 10^17 in 8*10^67 chance, it's 10^17 in 8*10^67/10^17. This is a massive difference."
It's true the odds scale down in 2 directions as the shuffles continue, massively reducing them and something I don't think all who watched QI grasped, but 10^17 in 8*10^67/10^17 isn't correct because it gives the impression that for each of the 10^17 shuffles there were 10^17 matches. Whilst that is correct for the 10^17th shuffle, it wasn't for the 1st, 2nd etc where there were only 1,2 etc matches to choose from. Recursion is needed for the proper equation.
Apologies for that (and the typos!).
I liked this post by the taliesin to try and picture just how many combinations we are talking about:
Start a timer that will count down the number of seconds from 52! to 0.
Start by picking your favorite spot on the equator. You're going to walk around the world along the equator, but take a very leisurely pace of one step every billion years. The equatorial circumference of the Earth is 40,075,017 meters.
After you complete your round the world trip, remove one drop of water from the Pacific Ocean. Now do the same thing again: walk around the world at one billion years per step, removing one drop of water from the Pacific Ocean each time you circle the globe. The Pacific Ocean contains 707.6 million cubic kilometers of water. Continue until the ocean is empty.
When it is, take one sheet of paper and place it flat on the ground. Now, fill the ocean back up and start the entire process all over again, adding a sheet of paper to the stack each time you've emptied the ocean.
Do this until the stack of paper reaches from the Earth to the Sun. Take a glance at the timer, you will see that the three left-most digits haven't even changed. You still have 8.063e67 more seconds to go. 1 Astronomical Unit, the distance from the Earth to the Sun, is defined as 149,597,870.691 kilometers. So, take the stack of papers down and do it all over again. One thousand times more. Unfortunately, that still won't do it. There are still more than 5.385e67 seconds remaining. You're just about a third of the way done.
Of course, in reality none of this could ever happen. The Pacific Ocean will boil off as the Sun becomes a red giant before you could even take your fifth step in your first trek around the world. Somewhat more of an obstacle, however, is the fact that all the stars in the universe will eventually burn out leaving space a dark, ever-expanding void inhabited by a few scattered elementary particles drifting a tiny fraction of a degree above absolute zero. The exact details are still a bit fuzzy, but according to some reckonings of The Reckoning, all this could happen before you would've had a chance to reduce the vast Pacific by the amount of a few backyard swimming pools.
http://czep.net/weblog/52cards.html
Now if that doesn't strike people as being impossible, I give up.
Start a timer that will count down the number of seconds from 52! to 0.
Start by picking your favorite spot on the equator. You're going to walk around the world along the equator, but take a very leisurely pace of one step every billion years. The equatorial circumference of the Earth is 40,075,017 meters.
After you complete your round the world trip, remove one drop of water from the Pacific Ocean. Now do the same thing again: walk around the world at one billion years per step, removing one drop of water from the Pacific Ocean each time you circle the globe. The Pacific Ocean contains 707.6 million cubic kilometers of water. Continue until the ocean is empty.
When it is, take one sheet of paper and place it flat on the ground. Now, fill the ocean back up and start the entire process all over again, adding a sheet of paper to the stack each time you've emptied the ocean.
Do this until the stack of paper reaches from the Earth to the Sun. Take a glance at the timer, you will see that the three left-most digits haven't even changed. You still have 8.063e67 more seconds to go. 1 Astronomical Unit, the distance from the Earth to the Sun, is defined as 149,597,870.691 kilometers. So, take the stack of papers down and do it all over again. One thousand times more. Unfortunately, that still won't do it. There are still more than 5.385e67 seconds remaining. You're just about a third of the way done.
Of course, in reality none of this could ever happen. The Pacific Ocean will boil off as the Sun becomes a red giant before you could even take your fifth step in your first trek around the world. Somewhat more of an obstacle, however, is the fact that all the stars in the universe will eventually burn out leaving space a dark, ever-expanding void inhabited by a few scattered elementary particles drifting a tiny fraction of a degree above absolute zero. The exact details are still a bit fuzzy, but according to some reckonings of The Reckoning, all this could happen before you would've had a chance to reduce the vast Pacific by the amount of a few backyard swimming pools.
http://czep.net/weblog/52cards.html
Now if that doesn't strike people as being impossible, I give up.
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