The theory of happiness. The curse of the director and the damned printers
I continue to acquaint the readers of Khabra with the chapters from my book The Theory of Happiness with the subtitle "Mathematical Foundations of the Laws of Meanness". This is not yet published popular science book, very informally telling about how mathematics allows a new degree of awareness to look at the world and people's lives. It is for those who are interested in science and for those who are interested in life. And since our life is complex and, by and large, unpredictable, the emphasis in the book is mainly on the theory of probability and mathematical statistics. Here theorems are not proved and the foundations of science are not given, it is by no means a textbook, but what is called recreational science. But it is this almost gaming approach that allows you to develop intuition, brighten up bright examples of lectures for students and, finally, explain to non-mathematicians and our children what we found so interesting in our dry science.
• Introduction to the Merfolology
• The law of the watermelon rind and the normality of the abnormality
• The law of a zebra and another's turn
We discuss the time pressures, deadlines and non-breaking printer.
"How to make a newspaper" and "How the play is staged" . Is the cause of this curse only in our disorganization and disorder? This, of course, the main reasons, but we are not so guilty in it that it was impossible to try to justify ourselves by some mathematical law. The strategy of the dunce, of course looks stupid, but the exponential growth rate is not a joke! Can I even cope with it?
The expected rate of work can be calculated accurately. The formula is not very elegant, but it is noteworthy that it includes the number of days
and does not include the number of scheduled cases:
The logarithm is a slow function, unless it is pressed against the wall. In the last days before the deadline, the tempo grows catastrophically, with the same speed with which the logarithm falls into the abyss as it approaches zero. However, from the number of days allocated, it still depends. You can look at what the expected pace for the week, month, and year looks like:
The most likely rate of performance of work in a limited time. It is interesting that the strict time limit affects beneficially. The name in stock is only a week, we are likely to do the work more evenly (half of the time will be ready for a third of the work), and if the whole year is ahead, then you can relax, well, and then regret it.
In an ideal perfectionist, who performs the work perfectly evenly, the pace of execution should tend to the diagonal (blue dotted line in the figure). This is similar to the equality curve on the Lorentz diagram, which signifies justice. Just as we calculated the Gini coefficient for the Lorentz diagram, we can, based on the area between the work rate curve and the ideal curve, calculate a certain meanness coefficient that will show how far we are from the ideal. It depends on the length of the allocated period and slowly increases with the growth of
. In the examples given for weeks, months and years, the coefficient of meanness is
How to deal with the growing wave of worries and time pressure? You can, for example, pull yourself together. A person with an excellent man's syndrome can strive to do the next thing as soon as possible, of course. A plausible model is the choice of the moment for the next case, following an exponential distribution with a density inversely proportional to the remaining time. This does not exclude some uncertainty inherent in our lives, but expresses the good intentions to do all things as soon as possible. Let's call this strategy good intentions strategy . That's what the distribution of probabilities for completing tasks on time for the adherent of this strategy, which in half the cases will do the next thing in the first quarter of the remaining time:
The distribution of probability does not have time for the strategy of good intentions.
Much better! Within a week, you can with a good chance to do five things and leave yourself two days off. But all the same, for longer periods, the increase in opportunities is not revolutionary. The problem lies in the fact that the expected number of successfully completed cases still remains proportional to the logarithm of the time released, and the logarithm grows extremely slowly! So, planning a lot, you need to keep in mind that the intensity of the process will inevitably increase, and the time ahead of the deadline is likely to be missed. In any case, it is necessary to remember that life is short and in order to realize the plan, it is necessary to act right now!
Let's admire the tempo of a well-meaning student.
The expected pace of work performance by a methodical person, trying to proceed to the next the stage of work as soon as possible. The graphs show the results of averaging of tens of thousands of numerical experiments simulating the execution of a task with a fixed number of stages. The red line indicates the maximum rate for a large number of tasks.
Our neatly managed to more evenly distribute the work, and do much more things, but he still expects time trouble. Short chains such a person will perform with a significant overfulfillment of the plan, and a chain of seven cases is almost perfect. However, as the number of cases increases, the expected rate rapidly tends to the theoretical tempo obtained with the help of the strategy of the dunce! The overall performance increased, but the parking before the deadline did not go away. So you can load up and load up a loader!
However, there is another well-known way to substantially discipline the performance of work: instead of one deadline, you have to make a lot of them. Let's break the deadline for doing the work in two equal parts and stick to this new deadline, considering it, say, an interim report. For each of these parts, we can plot the expected rate of work, as shown in the figure.
The breakdown of the work execution time into several intermediate reporting periods allows you to perform work more evenly, but adds stress at approach of each new report.
Despite the hassle with the interim report, we achieved our goal: the area under the general performance rate curve was reduced and the meanness ratio decreased from
. In addition, the reduction in the period (together with the reduction in the number of cases, of course) brings the expected rate of performance to an ideal, so the meanness has decreased more than twofold. Adding two more, say, quarterly reports, will reduce it to
, but by the same token, we drive our performers immediately into four stressful periods and they will still suffer loudly, complaining about fate and the authorities! Well, we can show the workers our calculations and prove that having introduced a quarterly report, they have lowered the coefficient of their meanness fivefold, if this, of course, will be a consolation to them.
Moreover, as the number of intermediate deadlines to the number of days allowed for work tends to increase, the pace of work will approach the ideal, but very boring pace.
Well! And the printer broke!
Add a few more words about the strategy of the dunce and the distribution of Stirling. The distribution obtained shows the probability of obtaining
events in a given time interval. Counting events in the present Poisson flow with the intensity
we arrive at the well-known Poisson distribution:
describing the credibility of obtaining
events in a single interval. The expression for Stirling numbers has an asymptotic expansion, which is larger than
reduces the distribution of the lengths of the chains with the deadline to the displaced Poisson distribution with the intensity
. Thus, our stochastic process with deadline, from the statistical point of view, can be considered either as a Poisson process on a condensing time grid or as an inhomogeneous Poisson process whose intensity monotonously and rapidly grows. And although, strictly speaking, our process is not Poissonian, since the events in it are not independent, however, the statistical properties we need are similar to them. Their similarity is indicated by the proximity of the average value and variance of the Stirling distribution characteristic for the Poisson distribution.
This conclusion allows us to ask the question: what if we add to our built-in process of doing a chain of cases, any rare troubles that are independent of us: a blizzard, a terrible cork, a runny nose, a printer breakdown, or a national holiday?
For the Poisson process, is defined. process of random thinning , which consists in the fact that we with some probability will remove events from the stream. Random decimation with probability
leaves the process Poisson, but its intensity decreases, multiplying by
. Events corresponding to a coincidence of trouble and some stage of the work themselves form a Poisson process, with a much lower intensity, but in our case, also, monotonously and rapidly growing. So swiftly that no matter how small the probability of trouble, for a sufficiently large number of cases (or the time allotted for work), it will be closer to the deadline to the fully observable. And the printer bets on the eve of the delivery of the course book!
Do not be surprised if the bus breaks down exactly when you are already late. The bus does not want you evil. Simply, if you are a girl, then the sequence of things: choose a dress, eat a candy, wash, put on your chosen dress, make up, put on a chain, move things from your purse to a clutch, clean your shoes and stuff, etc., to the most important and exciting deadline - to date !! And the pace with which you fly to meet fate is already so crazy that the most unlikely miracles begin to occur.
In the end, what is a miracle, how not the realization of the incredible!
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