Often, to make a single decision, they flip a coin, expecting to see a bird or a number. In rare cases, the coin will fall on the edge, confusing the “decisive” one.
Few people think that the use of a coin, such a yes / no method, is used even in mathematical experiments, and specifically in probability theory. Only in this case does the concept of a symmetric coin, sometimes called honest or mathematical, be used. This means that the density is the same throughout the coin, and the eagle or tails can fall out with the same probability. In addition to the familiar names of the parties, such a coin no longer has signs. No weight, no color, no size. Such a coin can give only two results - reverse or obverse, there are no "stand on the edge" in probability theory.
Everything in the world is likely
Probability theory is a whole field that is still trying to subjugate the case and calculate all the possible outcomes of events. Thanks to formulas and numerous empirical methods, this science makes it possible to judge the reasonableness of expectation. If we rely on the meaning of what Professor P. Laplace said (he made an important contribution to the development of the theory), then the essence of all actions in probability theory is an attempt to reduce the action of common sense to calculations.
The word "probably" directly refers to this science. The concept of “assumption” is used, which means: perhaps some event will occur. If you come closer to mathematics, then the most striking example is tossing a coin. And then we can assume: in a random experiment, a symmetric coin is thrown 100 times. It is likely that the emblem will be on top - from 45 to 55 times. Only then the assumption begins to be confirmed or proved by calculations.
Calculations against intuition
You can make a confirmation and turn to intuition. But what to do when the task is complicated? In practical experiments, more than one symmetric coin can be used. And then there are more options-combinations: two eagles, a tails and an eagle, two tails. The probability of falling out of each option is already different, and the combination “reverse - obverse” doubles in the loss compared to two eagles or two tails. The laws of nature will in any case be confirmed by physical experiments, and this situation can be similarly verified by tossing real coins.
There are situations when intuition is even more difficult to oppose to mathematical calculations. It is impossible to predict or feel all the options if there are even more coins. Mathematical tools related to combinatorial analysis are introduced.
Parsing Example
In a random experiment, a symmetric coin is flipped three times. It is necessary to calculate the probability of tails in all three throws.
The calculations. The tails should fall out in 100% of the cases of the experiment (3 times), this is one of 8 options-combinations: three eagles, two eagles and tails, etc. This means that the calculation of probability is done by dividing 100% by the total number of options. That is 1/8. We get an answer of 0.125.
There are plenty of tasks for a symmetric coin. But in probability theory there are examples that will interest even people who are far from mathematics.
sleeping Beauty
One of the paradoxes, the authorship of which is attributed to A. Elga, has a "fabulous" name. This very well reflects the essence of the paradox. This is a task that has several answers, and each of them is correct in its own way. The example clearly proves how easy it is to operate on the results using the most profitable result.
The sleeping beauty (the heroine of the experiment) is euthanized with sleeping pills through an injection. During this, a symmetric coin pops up. When the side with the eagle falls out, they awaken the heroine, ending the experiment. As a result, with tails, the beauty is awakened, after which they again euthanize it in order to wake up the next day of the experiment. At the same time, the beauty forgets that she was awakened, although she knew the conditions of the experiment, not counting the information on which day she woke up. Next is the most interesting question, specifically for an awakened beauty: "Calculate the probability of a side with a tails falling out."
There are two solutions to this paradoxical example.
In the first case, without proper information about the awakenings and the results of the loss of coins. Since a symmetric coin is involved, exactly 50% is obtained.
Second solution: for accurate data, the experiment is carried out 1000 times. It turns out that the beauty was woken up 500 times, if there was an eagle, and 1000 when tails. (Indeed, at the end of the tails, the heroine was asked twice). Accordingly, the probability is 2/3.
Vital
Similar data manipulation in statistics is found in life. For example, information about the share of pensioners in public transport. According to information, 40% of trips are made by pensioners. But in fact, pensioners do not make up 0.4 of the entire population. This is explained by the fact that retired people are more actively using transport services. Actually, the number of pensioners is recorded in the range of 18-20%. If we keep records of only the most recent passenger trip without taking into account the previous ones, the percentage of pensioners in the total passenger flow will be in the region of 20%. If you save all the data - then all 40%. It all depends on the subject using this data. Marketers need the first figure of the actual impressions of their advertising to the target audience, transporters are interested in the general figure.
It is noteworthy that from the mathematical layouts, something nevertheless leaked into real life. It was a symmetric coin that began to be used to resolve disputes due to its honest nature and the absence of any signs of partiality. For example, sports referees throw it when it is necessary to determine which of the participants will get the first move.