- Everything in the world is probable
- Calculating against intuition
- Example to parse
- Sleeping Beauty
Often, to make a single decision, a coin is tossed, expecting to see a bird or a number. In rare cases, the coin will fall on its edge, confusing the "decider".
Few people think that the use of a coin, a sort of "yes / no" method, is used even in mathematical experiments, and specifically in probability theory. Only in this case is the concept of a symmetrical coin sometimes called a fair or mathematical coin used. This means that the density is the same throughout the coin, and heads or tails can fall with the same probability. In addition to the names of the parties that have become familiar, such a coin no longer has any signs. No weight, no color, no size. Such a coin can give only two results - reverse or obverse, there are no "stand on edge" in probability theory.
Everything in the world is probable
Probability theory is a whole area that is still trying to subdue chance and calculate all possible outcomes of events. Thanks to formulas and numerous empirical methods, this science makes it possible to judgereasonable expectation. If we rely on the meaning of what was said by Professor P. Laplace (he made an important contribution to the development of the theory), then the essence of all actions in the theory of probability is an attempt to reduce the action of common sense to calculations.
The word "probably" refers directly to this science. The concept of "assumption" is used, which means: it is possible that some event will happen. If we come closer to mathematics, then the most striking example is tossing a coin. And then we can assume: in a random experiment, a symmetrical coin is tossed 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.
Calculating against intuition
You can make a counter-assertion and turn to intuition. But what to do when the task becomes more difficult? In practical experiments, more than one symmetrical coin can be used. And then there are more options-combinations: two eagles, tails and an eagle, two tails. The probability of falling out of each option becomes already different, and the combination "reverse - obverse" doubles in falling out 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 are introduced into the business,related to combinatorial analysis.
Example to parse
In a random experiment, a symmetrical coin is tossed three times. You need to calculate the probability of getting tails in all three throws.
Calculations. Tails must fall out in 100% of the cases of the experiment (3 times), this is one of 8 combinations: three heads, two heads and tails, etc. This means that the calculation of the probability is done by dividing 100% by the total number of options. That is 1/8. We get the answer 0, 125.
There are plenty of problems for a symmetrical coin. But there are examples in probability theory that will interest even people who are far from mathematics.
One of the paradoxes attributed to A. Elga has a "fabulous" name. This very well captures the essence of the paradox. This is a problem that has several answers, and each of them is correct in its own way. The example clearly shows how easy it is to operate on the results using the most profitable result.
Sleeping Beauty (the heroine of the experiment) is sedated with sleeping pills through an injection. During this, a symmetrical coin is tossed. When the side with the eagle falls out, the heroine is awakened, ending the experiment. With a result with tails, the beauty is awakened, after which they are again put to sleep in order to wake up the next day of the experiment. At the same time, the beauty forgets that she was awakened, although she knows the conditions of the experiment, not counting the information on which day she woke up. Next - the most interesting question, specifically for the awakened beauty: "Calculate the probability of getting a side with tails."
There are two solutions to this paradoxical example.
In the first case, without proper information about the wake-ups and the results of the coins. Since a symmetrical coin is involved, exactly 50% is obtained.
Second decision: for exact data, the experiment is carried out 1000 times. It turns out that the beauty was awakened 500 times if there was an eagle, and 1000 if it was tails. (After all, at the outcome with tails, the heroine was asked twice). Accordingly, the probability is 2/3.
Such manipulation of data in statistics occurs in life. For example, information on 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 total population. This is explained by the fact that retired people use transport services more actively. In reality, the number of pensioners is registered within 18-20%. If we take into account only the most recent passenger trip without taking into account the previous ones, then the percentage of pensioners in the total passenger traffic will be around 20%. If you save all the data, then all 40%. It all depends on the subject using this data. Marketers need the first digit of actual impressions of their ads to the target audience, transport workers are interested in the total number.
It is noteworthy that something from the mathematical layouts nevertheless leaked into real life. It was the symmetrical 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 refereesthey throw it up when it is necessary to determine which of the participants will get the first move.