Today is December 21st, winter solstice, the shortest day of the year, and (supposed) also on average the coldest one (even if this peculiarity is very variable belonging to the weather conditions). Today the sky is crystalline, there is a lazy sun (that woke up very late and soon will go back to sleep) and everything is shining with that golden yellow that match so good with the Christmas decorations.
Tomorrow, for some days, my wife, i and Maddie (our doggie) will go to spend Christmas in the mountain, with the intention to escape from the noise of the pagan festivity.
I love the silence in the mountain, where any word make sense because it does not confuse with all that background noise.
Too bad not to be next to our folks, a thought goes to my mom, my dad and my father-in-low. It would have been nice to spend Christmas with them, ending up sitting to chat in front of the softly crackling fireplace.
Friday, December 21, 2007
Wednesday, December 19, 2007
Pride.
Yesterday the United Nation General Assembly approved (104 to 54 votes, with 29 abstentions) the moratorium for abolishment of death penalty.
In this Victory of Civilization, Italian foreign politics took the lion's share, most passionate voice, in this campaign, among the European choir.
Impossible not to note that some of the opponents to the moratorium (Iran, North Korea, Sudan) are also the so-defined-by-the-United-States Rogue States, finding in this position a point of agreement with the United States theirselves.
Little the article that The New York Times gives to this fact, without mention the important Italian role.
As a "sad" Italian i am just amazed of how can some people be any "happy" during capital executions.
In this Victory of Civilization, Italian foreign politics took the lion's share, most passionate voice, in this campaign, among the European choir.
Impossible not to note that some of the opponents to the moratorium (Iran, North Korea, Sudan) are also the so-defined-by-the-United-States Rogue States, finding in this position a point of agreement with the United States theirselves.
Little the article that The New York Times gives to this fact, without mention the important Italian role.
As a "sad" Italian i am just amazed of how can some people be any "happy" during capital executions.
Friday, December 14, 2007
Italians? Sad people!
Really funny! Ian Fisher, correspondant from Rome of New York Times, wrote that Italians are the saddest people in Europe.
Reasonable that they point their finger against us because we have a system in which the services do not properly work, because our economy is not strong as the American one is, because we are full of contraddictions. But now.... we are also sad!
Maybe this Ian Fisher, expert on Italy of the prestigious newspaper, realized only now that we do not travel riding donkeys anymore, that Sofia Loren became an old lady, and that there's not a lot of us that spend our time playing mandulin and singing O Sole Mio. I would like to reassure anybody that read this blog with friendly attitude that i, Dario, am not sad at all.
Instead, if i were Ian Fisher, i would look into my own house and would notice that i belong to a Nation of invaders, that i have a dull President that drives a Nation against the interests of the citizens and the rest of the world, in order to favour the interests of the big multi-national companies that support him. I would try to understand how could it happen to voluntarily accept restrictions of personal freedoms and not to have any idea on how to have them back. Frankly, if i were Ian Fisher, i would ask myself if it is really appropriate to be so happy. Ian Fisher, if you are by chance reading, please accept this unbiased suggestion from a cheerful Italian. Go back to where you came from and have a laugh in a happy Country.
Reasonable that they point their finger against us because we have a system in which the services do not properly work, because our economy is not strong as the American one is, because we are full of contraddictions. But now.... we are also sad!
Maybe this Ian Fisher, expert on Italy of the prestigious newspaper, realized only now that we do not travel riding donkeys anymore, that Sofia Loren became an old lady, and that there's not a lot of us that spend our time playing mandulin and singing O Sole Mio. I would like to reassure anybody that read this blog with friendly attitude that i, Dario, am not sad at all.
Instead, if i were Ian Fisher, i would look into my own house and would notice that i belong to a Nation of invaders, that i have a dull President that drives a Nation against the interests of the citizens and the rest of the world, in order to favour the interests of the big multi-national companies that support him. I would try to understand how could it happen to voluntarily accept restrictions of personal freedoms and not to have any idea on how to have them back. Frankly, if i were Ian Fisher, i would ask myself if it is really appropriate to be so happy. Ian Fisher, if you are by chance reading, please accept this unbiased suggestion from a cheerful Italian. Go back to where you came from and have a laugh in a happy Country.
Wednesday, December 12, 2007
About determinism and tossing coins /2.
It is not really true that there is non-determinism (meaning "not a-priori determinable") about tossing a coin. On the contrary.
When i toss a coin i make so that from a position that it had with respect to the observers, it moves to another position. In order to do that i toss it in a certain way, expecting that it would fall about in a certain place, on one side or the other. In the previous post i gave some indication of things that can influence the motion, and so the result, bothering also mr. Coriolis. They were only examples, perhaps not even well hit, because i do not know if those causes really influence the motion, and even less, if they do, how much. But it is clear that the motion of the coin can be completely described by the external influences minutely formalizable. One can also ask if there is really any need to formalize them, instead of accepting the uncertainty of the result, but it is obvious that they are.
The reason we are sure they are formalizable can be found in Gödel's theorem about incompleteness of Arithmetics. In few words that theorem demostrates that a formal system (as Arithmetics is) can never be considered complete, which means that there is atleast one problem that does not fall in the formalization of that system, and so it is not solvable by mean of the rules of that system. Gödel's theorem so, says, between the lines, also other two things:
1) whatever can be expressed through that formal system, it is also formalizable, or it is possible to completely describe it with all the cause-effect relationships that insert it in a context.
2) that the formalization of a probelm outside the formal system presupposes then a formalization, which is the creation of a new formal system in which that new problem could be demonstrated.
In other words the Gödel theorem tells us that, observing the toss of a coin, i can ask myself "why does it fall on that side?", knowing that, even if we don't know the alswer of that question, that answer exists. In short, that question presupposes a search of a formalization of the events that influence the motion of the coin so that it falls on that side, but this search itself presupposes the hypotesis that the formalization exists.
In other words, answering to the question "why does it fall on that side?" with "there is no reason", not only we make the chain of cause-effect relationships vain, but also we make the question itself vain. Considering possible to pose oneself that question demanding that it makes sense, means trusting that, despite how difficult it could be, the answer exists. Who asks that question already knows (or believes) that the answer is deterministic.
So, it is not the chance that makes head or tail come out, in the sense that it is not a lack of rules that brings the result, unpredictable because given by a lack of rules. We can instead say that there are rules and they are well applied, but we do not have the technical capability to know them or, more simply, we don't feel like to investigate for them.
Then there is the problem that the observation can influence the production of the result.
Let's suppose that Werner Karl Heisenberg and i are in a closed room, sitting next to a table. Let's suppose that he has a coin in his hand and he tender to bet 100 euros on the result of the toss he's going to do. Let's suppose i bet on the side "head". The sound waves caused by my voice that pronounced that word generate a distribution of the molecula of air in the room that influences the motion of the coin so that it falls with result tail. If i was able to sharply compute this cause-effect relationship (or, in other words, to predict the behavior of all the variables involved, including the deterministic position of all the molecula of air in the room when i finish to pronounce the word "head", and so to predict that the result of the toss is tail), being motivated to win the bet, i wouldn't say "head", i would say "tail". But, my voice, pronouncing this word, generates a different distribution of the air molecula in the room, and such a distribution influence the coin in a different way, and the result is head. Being able to predict also this cause-effect relationship, the only chance to me is not to bet at all.
Obviously, this example is not very realistic. I am not able to find out if really the pronunciation of the word "tail" insted of "head" could influence on the motion of the coin enough to determine the result of the toss, because actually the study on this experiment would be too complex also to somebody much more clever than me. Moreover, and above all, the problem is that the experiment is not repeatable. If i pronounce the word "head" and the result is tail i could not propose Werner Karl to repeat the experiment with the same conditions, except my engagement to pronounce the word "tail". Because with all our good will, we wouldn't be able to recreate the same conditions. The coin would have been already tossed once. It would have fallen on the table, and as much the coin would have been clean there would have been some substances stuck on its surface which in part now moved on the table and on the Werner Karl's fingers, the air in the room moved and any molecula stopped in a different position. And even if we would also be able to put back all those elements in their original conditions, the neurons contained in my and my friend's brains theirselves would have different connections. If we were able to put back also the neurons to their original place, the result would be that we loose the memory of the previous experiment and the conclusions that we found, so we wouldn't remember about my engagement to say "tail".
And if we couldn't re-create exactly all the same conditions of the previous experiment, perhaps, pronouncing the word "tail" as established, finally the result would be tail. This would make me win the bet, but with the result to distort the experiment.
Actually it is unthinkable to consider everything (and with everything i mean really everything: what is included in the universe and the laws that control its behavior), because it is too complex. And so, it is better to rely to easier considerations accepting the fact that one cannot give a reliable answer to the toss-a-coin problem, but atleast it can be said that there is 50% of probability to win.
It is unthinkable to consider everything also for another reason, which is that the fact to think everything itself influence its behavior.
Let's put it in this way: let's suppose that one day human beings, even if not clever enough to consider the whole universe atom by atom and to determine so the behavior of any event, they are clever enough to build a super-super-computer with a super-super-software able to do it. This machine would be able of any forecast.
If i asked to that machine the result of the toss that Werner Karl is going to do, the machine would be able to tell me the answer. Even better, i wouldn't even need to ask it, because the machine would be able to forecast also my question. So, sitting in front of the terminal i would read the word "head" that is the answer to the question that i didn't ask but that the machine knows i would have. Knowing before the answer to my question, Werner Karl and i would know the result of the toss, which would make useless to really toss it. But if we don't toss the coin, asking the result of the toss would make no sense, therefore i wouldn't ask the machine the answer, and so the machine, forecasting it, wouldn't have any reason to give an answer. But if it happens in this way, Werner Karl and i would have an uncertainty about the result, we would decide then to procede with the toss, and we would like to know the result before tossing it, and so we would go back to the idea to ask the machine, and this is a vicious circle difficult to exit. Cool eh?
A machine like that would generate another logical problem. A program that consider a so huge amount of data would need the support of a huge hardware. How much hardware? Well... it's difficult to give an answer to this question because, let's suppose to fix at a certain point an amount that represent the number X of atoms to be considered. And let's suppose that that amount of atoms would be completely described by an amount of hardware made by a certain amount Y of atoms. Since the machine belongs to the universe too, and its behavior influence the behavior of the X atoms under analysis, at the end, to consider everything, the machine would have to describe also the behavior of the elements that constitute the machine itself, or, in other words, it has to work on an amount of data of X+Y atoms, and not X. But the hardware of this new model of machine cannot be made by a number Y of atoms, it must be made by a number a little bigger: let's say Y'. Which means that the complexive number of the atoms domain of the machine is not X+Y, but X+Y', which means that the hardware of the machine must be made by a number a little bigger of atoms, let's say Y''... In conclusion we should deduce that a machine capable to describe the whole universe must be atleast big as much as the universe is and so, its task restricts just to describe itself.
Actually we must say that a machine like that already exists. It is the universe itself, which is able to work and give the right results as effects of causes upon exact rules that work on exact datas. Infact we don't need a computer that tells us the result of the toss of a coin. It's enough that we toss it and look the result. If the result is head it means that the very complex calculation of that result generates the anwer "head", if it is tail, the answer "tail". And we are sure that the rules for which the result of the calculation would have been generated in that way have been perfectly respected, however complex they would be.
When i toss a coin i make so that from a position that it had with respect to the observers, it moves to another position. In order to do that i toss it in a certain way, expecting that it would fall about in a certain place, on one side or the other. In the previous post i gave some indication of things that can influence the motion, and so the result, bothering also mr. Coriolis. They were only examples, perhaps not even well hit, because i do not know if those causes really influence the motion, and even less, if they do, how much. But it is clear that the motion of the coin can be completely described by the external influences minutely formalizable. One can also ask if there is really any need to formalize them, instead of accepting the uncertainty of the result, but it is obvious that they are.
The reason we are sure they are formalizable can be found in Gödel's theorem about incompleteness of Arithmetics. In few words that theorem demostrates that a formal system (as Arithmetics is) can never be considered complete, which means that there is atleast one problem that does not fall in the formalization of that system, and so it is not solvable by mean of the rules of that system. Gödel's theorem so, says, between the lines, also other two things:
1) whatever can be expressed through that formal system, it is also formalizable, or it is possible to completely describe it with all the cause-effect relationships that insert it in a context.
2) that the formalization of a probelm outside the formal system presupposes then a formalization, which is the creation of a new formal system in which that new problem could be demonstrated.
In other words the Gödel theorem tells us that, observing the toss of a coin, i can ask myself "why does it fall on that side?", knowing that, even if we don't know the alswer of that question, that answer exists. In short, that question presupposes a search of a formalization of the events that influence the motion of the coin so that it falls on that side, but this search itself presupposes the hypotesis that the formalization exists.
In other words, answering to the question "why does it fall on that side?" with "there is no reason", not only we make the chain of cause-effect relationships vain, but also we make the question itself vain. Considering possible to pose oneself that question demanding that it makes sense, means trusting that, despite how difficult it could be, the answer exists. Who asks that question already knows (or believes) that the answer is deterministic.
So, it is not the chance that makes head or tail come out, in the sense that it is not a lack of rules that brings the result, unpredictable because given by a lack of rules. We can instead say that there are rules and they are well applied, but we do not have the technical capability to know them or, more simply, we don't feel like to investigate for them.
Then there is the problem that the observation can influence the production of the result.
Let's suppose that Werner Karl Heisenberg and i are in a closed room, sitting next to a table. Let's suppose that he has a coin in his hand and he tender to bet 100 euros on the result of the toss he's going to do. Let's suppose i bet on the side "head". The sound waves caused by my voice that pronounced that word generate a distribution of the molecula of air in the room that influences the motion of the coin so that it falls with result tail. If i was able to sharply compute this cause-effect relationship (or, in other words, to predict the behavior of all the variables involved, including the deterministic position of all the molecula of air in the room when i finish to pronounce the word "head", and so to predict that the result of the toss is tail), being motivated to win the bet, i wouldn't say "head", i would say "tail". But, my voice, pronouncing this word, generates a different distribution of the air molecula in the room, and such a distribution influence the coin in a different way, and the result is head. Being able to predict also this cause-effect relationship, the only chance to me is not to bet at all.
Obviously, this example is not very realistic. I am not able to find out if really the pronunciation of the word "tail" insted of "head" could influence on the motion of the coin enough to determine the result of the toss, because actually the study on this experiment would be too complex also to somebody much more clever than me. Moreover, and above all, the problem is that the experiment is not repeatable. If i pronounce the word "head" and the result is tail i could not propose Werner Karl to repeat the experiment with the same conditions, except my engagement to pronounce the word "tail". Because with all our good will, we wouldn't be able to recreate the same conditions. The coin would have been already tossed once. It would have fallen on the table, and as much the coin would have been clean there would have been some substances stuck on its surface which in part now moved on the table and on the Werner Karl's fingers, the air in the room moved and any molecula stopped in a different position. And even if we would also be able to put back all those elements in their original conditions, the neurons contained in my and my friend's brains theirselves would have different connections. If we were able to put back also the neurons to their original place, the result would be that we loose the memory of the previous experiment and the conclusions that we found, so we wouldn't remember about my engagement to say "tail".
And if we couldn't re-create exactly all the same conditions of the previous experiment, perhaps, pronouncing the word "tail" as established, finally the result would be tail. This would make me win the bet, but with the result to distort the experiment.
Actually it is unthinkable to consider everything (and with everything i mean really everything: what is included in the universe and the laws that control its behavior), because it is too complex. And so, it is better to rely to easier considerations accepting the fact that one cannot give a reliable answer to the toss-a-coin problem, but atleast it can be said that there is 50% of probability to win.
It is unthinkable to consider everything also for another reason, which is that the fact to think everything itself influence its behavior.
Let's put it in this way: let's suppose that one day human beings, even if not clever enough to consider the whole universe atom by atom and to determine so the behavior of any event, they are clever enough to build a super-super-computer with a super-super-software able to do it. This machine would be able of any forecast.
If i asked to that machine the result of the toss that Werner Karl is going to do, the machine would be able to tell me the answer. Even better, i wouldn't even need to ask it, because the machine would be able to forecast also my question. So, sitting in front of the terminal i would read the word "head" that is the answer to the question that i didn't ask but that the machine knows i would have. Knowing before the answer to my question, Werner Karl and i would know the result of the toss, which would make useless to really toss it. But if we don't toss the coin, asking the result of the toss would make no sense, therefore i wouldn't ask the machine the answer, and so the machine, forecasting it, wouldn't have any reason to give an answer. But if it happens in this way, Werner Karl and i would have an uncertainty about the result, we would decide then to procede with the toss, and we would like to know the result before tossing it, and so we would go back to the idea to ask the machine, and this is a vicious circle difficult to exit. Cool eh?
A machine like that would generate another logical problem. A program that consider a so huge amount of data would need the support of a huge hardware. How much hardware? Well... it's difficult to give an answer to this question because, let's suppose to fix at a certain point an amount that represent the number X of atoms to be considered. And let's suppose that that amount of atoms would be completely described by an amount of hardware made by a certain amount Y of atoms. Since the machine belongs to the universe too, and its behavior influence the behavior of the X atoms under analysis, at the end, to consider everything, the machine would have to describe also the behavior of the elements that constitute the machine itself, or, in other words, it has to work on an amount of data of X+Y atoms, and not X. But the hardware of this new model of machine cannot be made by a number Y of atoms, it must be made by a number a little bigger: let's say Y'. Which means that the complexive number of the atoms domain of the machine is not X+Y, but X+Y', which means that the hardware of the machine must be made by a number a little bigger of atoms, let's say Y''... In conclusion we should deduce that a machine capable to describe the whole universe must be atleast big as much as the universe is and so, its task restricts just to describe itself.
Actually we must say that a machine like that already exists. It is the universe itself, which is able to work and give the right results as effects of causes upon exact rules that work on exact datas. Infact we don't need a computer that tells us the result of the toss of a coin. It's enough that we toss it and look the result. If the result is head it means that the very complex calculation of that result generates the anwer "head", if it is tail, the answer "tail". And we are sure that the rules for which the result of the calculation would have been generated in that way have been perfectly respected, however complex they would be.
Tuesday, December 11, 2007
About determinism and tossing coins.
"Head or tail?"
"Head".
Tail.
I lost.
Probability science says that there is perfect uncertainity - non-determinism - in tossing a coin. And when there is perfect uncertainity, they say there is an equal probability for each one of the possible result. Since the sum must be 100%, then, there must be 50% for head and 50% for tail.
There are several consideration to say in order to parcially correct this hypotesis, for example, considering one particular coin that is going to be tossed, maybe the weight of one side is heavier than the other side, so that, when the coin is flying it likes better to fall with that side down, therefore there would be more probabilities that the result is the other side. Or we should take in consideration also the way the coin is tossed. For example i use to put the coin on the thumbnail of my right hand, and release it after pulling with the tip of my index. Some other people throw the coin some other way, and it can affect the path the coin is taking when it is flying.
Maybe we should mention also the speed and direction of the wind... And what about the Coriolis force?
The Coriolis force is something scientists use to justify strange effects that some events can have on the surface of the world. For example opening the hole of a bathtub full of water, a circular motion of water around the hole is produced, which is spinning always clockwise in the northern emisphere and counterclockwise in the southern. That is due to the fact that, as effect of the rotation of the Earth, in the northern emisphere something tied to the surface moves towards east if exposed to south faster then if exposed to north, since the circumference of the southern parallels is major of the circumference of the northern. To understand it, imagine to put a ring on the top of a ball. The most the ring is larger, the most it falls towards the "equator". If you put two rings and the ball is spinning, both of the rings make the same number of turns per minute, but the speed of a point on the side of the bigger one is faster than the speed on the side of the other. If one tries to walk straight on the north emisphere, then, he tends to turn right, or, in other words, to spin clockwise.
The force of Coriolis can influence the motion of a coin when it is flying after having been tossed. Or not?
Anyway, if we don't consider all those facts, which thing makes our life easier, we should say that since we have no idea if the coin is going to fall one or the other side, we conclude that there is equal probability for one or the other result. We don't know, so it can happen in 50% of cases. This is kind of weird!
"Composed probability" is the calculation of probability observing more than one event. Given two un-related events, the probability that one and the other happen is given by the probability that one happen multiplied by the probability that the other happen. On the opposite, the probability that one or the other happens is obviously given by 100% minus the probability that the first doesn't happen and the second doesn't either. In other words, given 1=100%, p1 the probability of event 1, p2 the one of event 2 we have:
probability of event 1 and event 2 =p1 * p2 ;
probability of event 1 or event 2 = 100% minus probability of not event 1 and not event 2, which is1 - (1 - p1) * (1 - p2) .
As an example, let's calculate the probability that tossing two coins we got at least one head. The cases are:
- two heads, winning
- the first head and the second tail, winning
- the first tail and the second head, winning
- two tails, loosing.
We win 3 times on four, which is 75%, infact the probability the first is not head is 50% (or 0.5), the probability that the second is not head is 50% too, so, the probability of one or the other being head is1 - 0.5 * 0.5 , which makes 1 - 0.25 = 0.75 = 75% .
Another way to consider the problem is this: we toss the first coin. If the result is head (50%), we win, and we don't need to toss the second. If the result is tail (50%) we have 50% of the remaining probabilities (which is 25% of the total) that it comes out head, so we win for the50% + 25% = 75% of cases.
Extending the calculation to a higher number of events, the probability that atleast one happens is given by:
1 - (1 - p1) * (1 - p2) * (1 - p3) * ....
For example, tossing 10 coins, the probability that atleast one is head is given by:
1 - 0.5 * 0.5 * ... 10 times , or 1 - 0.510 = 1 - 0.0009765625 = 0.9990234375 = 99.90234375% . Almost 100%!
Going a little further, if we toss 100 coins, the probability is 99.999999999999999999999999999921%. Not bad, uh?
We said that when there is something we don't know if can happen, there is 50% of probabilities it can. The funny conclusion of this is in the following example.
Let's imagine one planet in another star system. We don't know anything about that planet except the fact that it exist. The question is: do elephants exist on that planet? we have no idea! So we must say that the probabilities are 50% (being that the other 50% are the probabilities elephants do not exist). And what about whales? Same thing, in that planet there can be whales with probability 50%. So, how many probabilities there are that elephants or whales exist in that planet? Just like tossing two coins: 75%. And what about mices? and crikets? and flees? Human beings? Not to count tomatoes or apples or bacteria of Antrax. If we take in consideration only 100 different forms of life, the probabilities that atleast one of them can be found on that planet are therefore 99.999999999999999999999999999921%.
In conclusion, if we take in consideration one planet of which we don't know anything at all, there are almost 100% of probabilities that there is atleast a form of life. Would you believe it? I wouldn't.
One explaination of this paradox is that it's not real that we don't really know nothing at all about that planet. For example we know the type of material it is made of (atoms) because we believe that the whole universe is made in that way, we know that it can be gasiform, in which case life cannot exist as we know it, or it can be big or little, depending on which force of gravity can be too strong and smashing any kind of life or too weak to keep gases attracted and so to have an atmosphere in which life can breathe. We know that, to allow life, the chemistry of that planet must allow the formation of molecula on a certain atomic model and blah blah blah... well... i am not a scientist so clever to know all those details, but what i want to say is that it's not real that we don't know anything at all about that planet. Infact what we know is that the conditions to find a form of life on a planet are really difficult to have. We know that.
But if we didn't know it, would we conclude that there would be almost for sure some kind of life on any unknown planet?
The wrong thing is that we give to an event a lot of chances to happen even if we don't know anything at all about that event. Rather, right because we don't! If we didn't know anything about coins, would we bet that, tossing one 100 times it would result head about 50 times (whatever we mean with "about")? The fact is that we know a lot about coins. And if we don't believe, we could ourself try to toss one a big number of times and count how many times it results head. One important thing we know, for example, is that the coin has two sides, and it will fall on the table on one side or the other. This simple rule for example does not apply on a spaceship where there's no gravity, for the simple reason that there is not an "up" where to toss the coin nor a "down" where the coin is going to fall.
"Head".
Tail.
I lost.
Probability science says that there is perfect uncertainity - non-determinism - in tossing a coin. And when there is perfect uncertainity, they say there is an equal probability for each one of the possible result. Since the sum must be 100%, then, there must be 50% for head and 50% for tail.
There are several consideration to say in order to parcially correct this hypotesis, for example, considering one particular coin that is going to be tossed, maybe the weight of one side is heavier than the other side, so that, when the coin is flying it likes better to fall with that side down, therefore there would be more probabilities that the result is the other side. Or we should take in consideration also the way the coin is tossed. For example i use to put the coin on the thumbnail of my right hand, and release it after pulling with the tip of my index. Some other people throw the coin some other way, and it can affect the path the coin is taking when it is flying.
Maybe we should mention also the speed and direction of the wind... And what about the Coriolis force?
The Coriolis force is something scientists use to justify strange effects that some events can have on the surface of the world. For example opening the hole of a bathtub full of water, a circular motion of water around the hole is produced, which is spinning always clockwise in the northern emisphere and counterclockwise in the southern. That is due to the fact that, as effect of the rotation of the Earth, in the northern emisphere something tied to the surface moves towards east if exposed to south faster then if exposed to north, since the circumference of the southern parallels is major of the circumference of the northern. To understand it, imagine to put a ring on the top of a ball. The most the ring is larger, the most it falls towards the "equator". If you put two rings and the ball is spinning, both of the rings make the same number of turns per minute, but the speed of a point on the side of the bigger one is faster than the speed on the side of the other. If one tries to walk straight on the north emisphere, then, he tends to turn right, or, in other words, to spin clockwise.
The force of Coriolis can influence the motion of a coin when it is flying after having been tossed. Or not?
Anyway, if we don't consider all those facts, which thing makes our life easier, we should say that since we have no idea if the coin is going to fall one or the other side, we conclude that there is equal probability for one or the other result. We don't know, so it can happen in 50% of cases. This is kind of weird!
"Composed probability" is the calculation of probability observing more than one event. Given two un-related events, the probability that one and the other happen is given by the probability that one happen multiplied by the probability that the other happen. On the opposite, the probability that one or the other happens is obviously given by 100% minus the probability that the first doesn't happen and the second doesn't either. In other words, given 1=100%, p1 the probability of event 1, p2 the one of event 2 we have:
probability of event 1 and event 2 =
probability of event 1 or event 2 = 100% minus probability of not event 1 and not event 2, which is
As an example, let's calculate the probability that tossing two coins we got at least one head. The cases are:
- two heads, winning
- the first head and the second tail, winning
- the first tail and the second head, winning
- two tails, loosing.
We win 3 times on four, which is 75%, infact the probability the first is not head is 50% (or 0.5), the probability that the second is not head is 50% too, so, the probability of one or the other being head is
Another way to consider the problem is this: we toss the first coin. If the result is head (50%), we win, and we don't need to toss the second. If the result is tail (50%) we have 50% of the remaining probabilities (which is 25% of the total) that it comes out head, so we win for the
Extending the calculation to a higher number of events, the probability that atleast one happens is given by:
For example, tossing 10 coins, the probability that atleast one is head is given by:
Going a little further, if we toss 100 coins, the probability is 99.999999999999999999999999999921%. Not bad, uh?
We said that when there is something we don't know if can happen, there is 50% of probabilities it can. The funny conclusion of this is in the following example.
Let's imagine one planet in another star system. We don't know anything about that planet except the fact that it exist. The question is: do elephants exist on that planet? we have no idea! So we must say that the probabilities are 50% (being that the other 50% are the probabilities elephants do not exist). And what about whales? Same thing, in that planet there can be whales with probability 50%. So, how many probabilities there are that elephants or whales exist in that planet? Just like tossing two coins: 75%. And what about mices? and crikets? and flees? Human beings? Not to count tomatoes or apples or bacteria of Antrax. If we take in consideration only 100 different forms of life, the probabilities that atleast one of them can be found on that planet are therefore 99.999999999999999999999999999921%.
In conclusion, if we take in consideration one planet of which we don't know anything at all, there are almost 100% of probabilities that there is atleast a form of life. Would you believe it? I wouldn't.
One explaination of this paradox is that it's not real that we don't really know nothing at all about that planet. For example we know the type of material it is made of (atoms) because we believe that the whole universe is made in that way, we know that it can be gasiform, in which case life cannot exist as we know it, or it can be big or little, depending on which force of gravity can be too strong and smashing any kind of life or too weak to keep gases attracted and so to have an atmosphere in which life can breathe. We know that, to allow life, the chemistry of that planet must allow the formation of molecula on a certain atomic model and blah blah blah... well... i am not a scientist so clever to know all those details, but what i want to say is that it's not real that we don't know anything at all about that planet. Infact what we know is that the conditions to find a form of life on a planet are really difficult to have. We know that.
But if we didn't know it, would we conclude that there would be almost for sure some kind of life on any unknown planet?
The wrong thing is that we give to an event a lot of chances to happen even if we don't know anything at all about that event. Rather, right because we don't! If we didn't know anything about coins, would we bet that, tossing one 100 times it would result head about 50 times (whatever we mean with "about")? The fact is that we know a lot about coins. And if we don't believe, we could ourself try to toss one a big number of times and count how many times it results head. One important thing we know, for example, is that the coin has two sides, and it will fall on the table on one side or the other. This simple rule for example does not apply on a spaceship where there's no gravity, for the simple reason that there is not an "up" where to toss the coin nor a "down" where the coin is going to fall.
Monday, December 10, 2007
I start (again) from here.
I decided to try (again) to keep a blog active, writing something every now and then... just anything it happens i'd like to say. I hope i am not going to be boring too much.
If you ever pass by, and you feel like to leave a comment, it will make me happy.
I also decided to be completely bilingual, infact there is another blog belonging to a parallel universe over here, identical but in another language
If you ever pass by, and you feel like to leave a comment, it will make me happy.
I also decided to be completely bilingual, infact there is another blog belonging to a parallel universe over here, identical but in another language
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