"What the world can not be at all ..."
Dear friends, I am pleased to present you another article from the series of my travels around the world of amazing. We began by talking about giant numbers , where I tried to share with you my admiration of what incredible numbers surround us in the Universe and how close we can go to infinity itself. The second article told about microscopically small objects that are far beyond the visibility of not only the naked eye, but also the strongest microscope. Now I suggest you go on a third journey - a journey into the world of probabilities. We will consider examples of incredible, but, nevertheless, mathematically possible events. We will have to work with numbers again, so I apologize to the humanities in advance (unless, of course, there are any on this resource). In general, if you, like me, love to hammer in your head with useless facts, then welcome.
Let us briefly touch upon the question of what probability is. This term is often misunderstood. Probability is a quantitative assessment of the possibility of the occurrence of a certain event. It can take values ​​from zero to one. A common mistake that is made here is that this assessment is made after the occurrence (or non-occurrence) of the event. Although in this case, the probability as if “collapses” and takes only one of two values: 0 or 1. That is, the event or has come or not, and no calculations have nothing to do with it. A good illustration of this statement will be a famous saying about the futility of waving fists after a fight.
Obviously, events are more and less likely. I suggest to go on a journey to those whose probability is so close to zero that it is already breathtaking. Of course, for this we need some knowledge about the world, the ability to count and imagination. I'm not sure that I have something else, or even a third one, but still let's get started.
')
And we start with a probability of 0.5. These are the chances to meet on the street dinosaur ("or meeting, or not meeting") - a well-known joke. But seriously, one of the most popular event providers with this probability is an ordinary coin. Quite often, decisions are made on the basis of whether it will fall on tails or on tails. For example, in 1968, at the European Football Championship, the semifinal meeting between the USSR and Italy teams ended in a draw, and its outcome was decided by a coin flip. By the way, the Italians reached the final (and even became European champions), while ours were left without medals, losing in the match for 3rd place. This is the cunning probability of 0.5.
However, even if the coin is “ideal” and even if we throw it also “ideally” (which in the real world is, of course, unattainable), the probability that it will fall on heads or tails will not be strictly equal to 0.5. A coin can be on the edge - and this is quite a possible situation, the probability of which strongly depends on the thickness of the coin and the method of throw. For a more or less standard coin, quite roughly it can be rated as “1 to 6 000” , which, you see, is not at all so little. For example, if you make a million shots, you can expect the coin to stand on the edge about 150 times. That is, it will happen about 1 time in 2 days, if you throw a whole year for 8 hours each day. But if you want to wait for the coin to stand on the edge two times in a row, then you will have to throw coins at the same pace for about 35 years. In principle, for very persistent people, this is also real.
Of course, “one to two” or even “one to six thousand” is not interesting. We face such events all the time. I am sure that you were waiting for something more, so let's not waste time and immediately make a leap to the “one in a million” probability. What events occur with such or about such a probability? This is the chance to win the top prize in any lottery. For example, the chance to guess 6 numbers from 45 is equal to “1 to 8 145 060”. It is worth mentioning the funny episode that occurred in September 2009 in Bulgaria, when in the lottery “6 out of 42” identical winning numbers fell out in two runs in a row (the article indicates the probability of this as “1 to 4.2 million”, but personally I got another the value is "1 to 5 245 786"). You can also try to throw 9 playing dice so that all of them have the same value - “1 to 1 679 616”. Or throw out 20 eagles from 20 abandoned coins - “1 to 1 048 576”. The probability to die from a tsunami is “1 to 500 000”. Chances to get a “flash royal” in poker are also approximately in this range - “1 to 649 740”.
In general, in order for an event with a probability of “one in a million” to occur at least once in a lifetime, you need to “try” 50 times every day. For example, if we intersect with 50 random people every day, then sometime in our life we ​​will encounter a chance of meeting with which is “one in a million.” In principle, this value is close to one in which you can safely not believe. Although if we consider that more than 7 billion people live on the globe, even such rare events happen to someone every day.
Moving on, to even more incredible events, to wait for the onset of which, during the course of their own lives, there is definitely no point. The probability of a global catastrophe for civilization as a result of the fall of a large asteroid in a given year is “1 in 10 million” (however, for one particular person the chance to die from the asteroid is 1 in 500 thousand, so be careful). The probability of the first attempt to find a needle in a haystack is approximately “1 to 100 million” (and if not with the first one and under certain conditions, then it can be managed in just 2 days ). Is it possible to randomly assemble a disassembled Rubik's cube 3x3x3? Any particular arrangement has a chance to be correct only "1 to 4.32 * 10 19 ", so the assembly may take 26 years .
The probability of firing from an infinitely thin laser (these do not happen) to an arbitrary point of the sky and getting into Voyager-1 is “1 to 5 * 10 26 ” (to the Sun or the Moon - “1 to 180 000”). And if you just poke at a random point in the Universe, then you get into something with a probability of just “1 to 10 31. ” The probability of coincidence of fingerprints of two different people is “1 to 10 60 ”. A person who does not know how to play chess has a chance to win about 1 to 10 100 from the grandmaster, if we assume that he will have to guess about 40 moves, each time making one of two or three suitable of the 500 available in total (including that do not comply with the rules).
Let's make one more qualitative leap to complete improbability. The probability that by pressing keys on the keyboard at random, we will type the exact text of this article (about 11,000 characters in it) equal to "1 to 40 11000 ". This is a number in which, after zero and a comma, another 17,622 zeros follow, and only then some significant digits. This is an extremely low probability, so you can be sure that I’m working hard to write something meaningful, and not just randomly knocking on the keyboard.
If we take as an illustration an example that has already become a certain standard, then I remind you that the probability of randomly typing the original text of "Hamlet" is approximately "1 in 10 183946 ". This number is zero in any practical sense, and it will not grow any tangible, even if we fill the entire Universe with Planck-size typewriters and wait for a time comparable to the time of its life. The problem with these numbers is that they are so small that they simply have nothing to compare them with, except for others that are just as unreal. Moreover, it is not the numbers themselves that have to be compared, but the exponents in their denominators. And let me remind you that even if this indicator is 6, then such an event will almost certainly not happen during the life of one person.
The chances for a man thrown out into space to be picked up by a randomly flying starship for 30 seconds are known to be “1 to 2 267709 ” or “1 to 10 80588 ” (“Hitchhiker's Guide to the Galaxy”, D. Adams). Another popular example is the probability that all the air in a room in which you are located will randomly gather in one of its halves. Mathematically, this is possible, and there are no physical laws that forbid such distribution to molecules. The probability of this for a room with a volume of 40 cubic meters is approximately equal to “1 to 2 10 27 ” (in the denominator, we have two in the octillion degree here - is that impressive?) Or “1 to 10 10 26.5 ”. This number is much closer to zero than any of the previous ones. In order to somehow work with it, you have to use the wonderful WolframAlpha calculator .
And one more example in pursuit. Let us draw a 10-liter bucket of water from the lake. Can it be so that all the water in it will be "heavy" (in which instead of hydrogen there will be deuterium - its isotope with atomic mass equal to 2)? For reference: the content of heavy water in normal - 1 molecule per 5 500, and in a bucket of water about 10 27 molecules. We get the probability: "1 to 5500 10 27 " or "1 to 10 10 27.5 ". Please note that in the previous example, we raised a power of two, and in this “5,500,” and nevertheless obtained similar results. This happened because the exponents we had were breathtakingly high. To the power of 10 27 , in principle, it doesn’t matter what number to build (if it is at least a little more than 1) - the results will be equally huge.
We are already close to completion, so we will strain our imagination for the last time and imagine that we have disassembled the entire visible Universe into Planck volume cubes (suppose they all turned out to be different), then they should be mixed and re-arranged in a random order. What is the probability that they will each take their place? It is equal to "1 to 10 185 !" Or "1 to 10 10 187 ". I think that this number is already close enough to zero to stop at it.
If you have examples of any amazing events or coincidences, I will be grateful if you write about them in the comments. And also, as usual, I would like to ask dear readers to point out errors and inaccuracies in this text. From my previous texts you most likely understood that I am not a physicist, and I suppose that now suspicions creep in on you that I am not a mathematician either. It's true. So there must be mistakes. I would like to correct them and learn something.
Let us briefly touch upon the question of what probability is. This term is often misunderstood. Probability is a quantitative assessment of the possibility of the occurrence of a certain event. It can take values ​​from zero to one. A common mistake that is made here is that this assessment is made after the occurrence (or non-occurrence) of the event. Although in this case, the probability as if “collapses” and takes only one of two values: 0 or 1. That is, the event or has come or not, and no calculations have nothing to do with it. A good illustration of this statement will be a famous saying about the futility of waving fists after a fight.
Obviously, events are more and less likely. I suggest to go on a journey to those whose probability is so close to zero that it is already breathtaking. Of course, for this we need some knowledge about the world, the ability to count and imagination. I'm not sure that I have something else, or even a third one, but still let's get started.
')
And we start with a probability of 0.5. These are the chances to meet on the street dinosaur ("or meeting, or not meeting") - a well-known joke. But seriously, one of the most popular event providers with this probability is an ordinary coin. Quite often, decisions are made on the basis of whether it will fall on tails or on tails. For example, in 1968, at the European Football Championship, the semifinal meeting between the USSR and Italy teams ended in a draw, and its outcome was decided by a coin flip. By the way, the Italians reached the final (and even became European champions), while ours were left without medals, losing in the match for 3rd place. This is the cunning probability of 0.5.
However, even if the coin is “ideal” and even if we throw it also “ideally” (which in the real world is, of course, unattainable), the probability that it will fall on heads or tails will not be strictly equal to 0.5. A coin can be on the edge - and this is quite a possible situation, the probability of which strongly depends on the thickness of the coin and the method of throw. For a more or less standard coin, quite roughly it can be rated as “1 to 6 000” , which, you see, is not at all so little. For example, if you make a million shots, you can expect the coin to stand on the edge about 150 times. That is, it will happen about 1 time in 2 days, if you throw a whole year for 8 hours each day. But if you want to wait for the coin to stand on the edge two times in a row, then you will have to throw coins at the same pace for about 35 years. In principle, for very persistent people, this is also real.
Of course, “one to two” or even “one to six thousand” is not interesting. We face such events all the time. I am sure that you were waiting for something more, so let's not waste time and immediately make a leap to the “one in a million” probability. What events occur with such or about such a probability? This is the chance to win the top prize in any lottery. For example, the chance to guess 6 numbers from 45 is equal to “1 to 8 145 060”. It is worth mentioning the funny episode that occurred in September 2009 in Bulgaria, when in the lottery “6 out of 42” identical winning numbers fell out in two runs in a row (the article indicates the probability of this as “1 to 4.2 million”, but personally I got another the value is "1 to 5 245 786"). You can also try to throw 9 playing dice so that all of them have the same value - “1 to 1 679 616”. Or throw out 20 eagles from 20 abandoned coins - “1 to 1 048 576”. The probability to die from a tsunami is “1 to 500 000”. Chances to get a “flash royal” in poker are also approximately in this range - “1 to 649 740”.
In general, in order for an event with a probability of “one in a million” to occur at least once in a lifetime, you need to “try” 50 times every day. For example, if we intersect with 50 random people every day, then sometime in our life we ​​will encounter a chance of meeting with which is “one in a million.” In principle, this value is close to one in which you can safely not believe. Although if we consider that more than 7 billion people live on the globe, even such rare events happen to someone every day.
Moving on, to even more incredible events, to wait for the onset of which, during the course of their own lives, there is definitely no point. The probability of a global catastrophe for civilization as a result of the fall of a large asteroid in a given year is “1 in 10 million” (however, for one particular person the chance to die from the asteroid is 1 in 500 thousand, so be careful). The probability of the first attempt to find a needle in a haystack is approximately “1 to 100 million” (and if not with the first one and under certain conditions, then it can be managed in just 2 days ). Is it possible to randomly assemble a disassembled Rubik's cube 3x3x3? Any particular arrangement has a chance to be correct only "1 to 4.32 * 10 19 ", so the assembly may take 26 years .
The probability of firing from an infinitely thin laser (these do not happen) to an arbitrary point of the sky and getting into Voyager-1 is “1 to 5 * 10 26 ” (to the Sun or the Moon - “1 to 180 000”). And if you just poke at a random point in the Universe, then you get into something with a probability of just “1 to 10 31. ” The probability of coincidence of fingerprints of two different people is “1 to 10 60 ”. A person who does not know how to play chess has a chance to win about 1 to 10 100 from the grandmaster, if we assume that he will have to guess about 40 moves, each time making one of two or three suitable of the 500 available in total (including that do not comply with the rules).
Let's make one more qualitative leap to complete improbability. The probability that by pressing keys on the keyboard at random, we will type the exact text of this article (about 11,000 characters in it) equal to "1 to 40 11000 ". This is a number in which, after zero and a comma, another 17,622 zeros follow, and only then some significant digits. This is an extremely low probability, so you can be sure that I’m working hard to write something meaningful, and not just randomly knocking on the keyboard.
If we take as an illustration an example that has already become a certain standard, then I remind you that the probability of randomly typing the original text of "Hamlet" is approximately "1 in 10 183946 ". This number is zero in any practical sense, and it will not grow any tangible, even if we fill the entire Universe with Planck-size typewriters and wait for a time comparable to the time of its life. The problem with these numbers is that they are so small that they simply have nothing to compare them with, except for others that are just as unreal. Moreover, it is not the numbers themselves that have to be compared, but the exponents in their denominators. And let me remind you that even if this indicator is 6, then such an event will almost certainly not happen during the life of one person.
The chances for a man thrown out into space to be picked up by a randomly flying starship for 30 seconds are known to be “1 to 2 267709 ” or “1 to 10 80588 ” (“Hitchhiker's Guide to the Galaxy”, D. Adams). Another popular example is the probability that all the air in a room in which you are located will randomly gather in one of its halves. Mathematically, this is possible, and there are no physical laws that forbid such distribution to molecules. The probability of this for a room with a volume of 40 cubic meters is approximately equal to “1 to 2 10 27 ” (in the denominator, we have two in the octillion degree here - is that impressive?) Or “1 to 10 10 26.5 ”. This number is much closer to zero than any of the previous ones. In order to somehow work with it, you have to use the wonderful WolframAlpha calculator .
And one more example in pursuit. Let us draw a 10-liter bucket of water from the lake. Can it be so that all the water in it will be "heavy" (in which instead of hydrogen there will be deuterium - its isotope with atomic mass equal to 2)? For reference: the content of heavy water in normal - 1 molecule per 5 500, and in a bucket of water about 10 27 molecules. We get the probability: "1 to 5500 10 27 " or "1 to 10 10 27.5 ". Please note that in the previous example, we raised a power of two, and in this “5,500,” and nevertheless obtained similar results. This happened because the exponents we had were breathtakingly high. To the power of 10 27 , in principle, it doesn’t matter what number to build (if it is at least a little more than 1) - the results will be equally huge.
We are already close to completion, so we will strain our imagination for the last time and imagine that we have disassembled the entire visible Universe into Planck volume cubes (suppose they all turned out to be different), then they should be mixed and re-arranged in a random order. What is the probability that they will each take their place? It is equal to "1 to 10 185 !" Or "1 to 10 10 187 ". I think that this number is already close enough to zero to stop at it.
If you have examples of any amazing events or coincidences, I will be grateful if you write about them in the comments. And also, as usual, I would like to ask dear readers to point out errors and inaccuracies in this text. From my previous texts you most likely understood that I am not a physicist, and I suppose that now suspicions creep in on you that I am not a mathematician either. It's true. So there must be mistakes. I would like to correct them and learn something.
Source: https://habr.com/ru/post/365855/
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