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My book writes:

From experience it has been observed that the value of frequency ratio gradually approaches a definite constant number when the no. of trial becomes larger & larger. This phenomenon of stability of frequency ratio when a random experiment is repeated under indentical conditions a large no. of times is called Statistical Regularity.

Now,what is the cause for the so-called stability of the frequency-ratio? What does this want to tell??

Let a coin is tossed $30$ times. Now, $15$ is our estimate for the number of heads. But after performing it many times,hardly one time head appeared for $15$ times. So,after seeing it , one will estimate the probability for heads in the next flip is $0.448$ not $0.5$ . Now,what magic does happen that makes the probability $0.5$ when the experiment is done infinitely? This means, when the number of experiment,here tossing of the coin, done is finite, then the head comes less than half of the number of experiment. But when the number of experiment becomes infinite, then the favoured event exactly comes half of the number of experiment. Why? What is the cause??

  • The magic is that when you toss the coin many times, the factors that influence the result, small movements of the hand, air currents, etc. will appear an equal number of times so as to produce one value, and so as to produce the other value. – Sofia Jan 11 '15 at 13:00
  • @Sofia: Ma'am , can you please explain me the magic with any other example? –  Jan 11 '15 at 13:25
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    (no need for ma'am, we are all friends here) Yes, of course I can. Another example is to cast a cube on which surfaces are painted in 6 different colors. If the cube is perfectly homogenous, the probability that the cube falls with the red surface up is 1/6. In fact, probabilities are not necessary in calculating which face will be up in a given cast. If a mechanic hand is used, which does with precision the same movement, if the floor is perfectly smooth, if there are no air currents, the cube will always fall in the same way. – Sofia Jan 11 '15 at 18:15
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    The probabilities appear because we can't control all these factors. However, if these factors are "fair", i.e. their "values" that produce red up, and yellow up, and green up, etc., appear with equal frequency, the six colors will appear with equal frequencies. Now, by the "values" I mean that once you keep your hand with which you cast the cube higher, once lower, once more turned to the left, etc. – Sofia Jan 11 '15 at 18:23
  • The probabilities appear because we can't control all these factors. However, if these factors are "fair", i.e. their "values" that produce red up, and yellow up, and green up, etc., appear with equal frequency, the six colors will appear with equal frequencies. Now, by the "values" I mean that once you keep your hand with which you cast the cube higher, once lower, once more turned to the left, etc. If you want additional examples, you can calculate the probability to get a mail. If you know all the people that may write to you, and what they have to do every day, – Sofia Jan 11 '15 at 18:29
  • (cont.) then you can say with certainty if in given day you will get a mail. But if not, the probability to get is the same as for no mail, i.e. 1/2. – Sofia Jan 11 '15 at 18:31
  • Thanks for your help. So,the factors that influence the event appear for such a number that the probability becomes equal to that of theoretical one . Right? –  Jan 12 '15 at 06:35
  • If an experiment is done,how can there be any probability? I've asked a quo. If you help, I'll be grateful. Thanks. –  Jan 12 '15 at 07:52
  • if the factors that influence the experiment result have an absolutely randomal character, e.g. the air currents (that can influence how the cube falls) go once in one direction, another time in another direction, and all the directions are equally probable, then, the cube will fall on each side with equal probability. Now I look at the quo. – Sofia Jan 12 '15 at 11:30
  • Well, I would do the things otherwise than you say. After 2 trials I would calculate the ratio $m/n$, i.e. $n = 2$ and $m=$ the number of times A appeared. So, I would obtain $P_2(A)$. One more trial, and I would calculate $P_3(A)$, and so on, until $P_{500}(A)$. Then, I would estimate to which value converges the series of probabilities. If you repeat the experiment 500 times more, and you see that the series of frequencies establishes on 0.448, not on 0.5, then your coin is slightly uneven, or, the conditions of the experiment are slightly not even. – Sofia Jan 12 '15 at 11:45

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