I am wondering if we assume "identical flips" when we perform a series of coin toss experiments to get the probability of either heads of tails. The way I understand "identical flips" is that it means each coin toss is done exactly in the same environments although each coin toss is not really experimented each time in the same environments(forces on the coin, angle, distance, and etc) but we just assume each coin toss is done in the same environment.

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    $\begingroup$ We often assume sample flips are independent and identically distributed or i.i.d. for short. So you should usually not get all the observations identical, though the belief is that the distributions are identical. $\endgroup$
    – Henry
    Nov 9 '20 at 0:02
  • $\begingroup$ Does "identically distributed" mean that I toss the coin exactly in the same manner like the previous ones? $\endgroup$
    – StoryMay
    Nov 9 '20 at 6:14
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    $\begingroup$ How could two coin tosses be "identical" if one came up tails and the other came up heads? Doesn't randomness impose the possibility that the tosses are not identical? $\endgroup$ Nov 9 '20 at 7:41
  • $\begingroup$ We assume data are generated in the same data generation process but observed data have different values according to randomness. This is how I understand about the identical flips. $\endgroup$
    – StoryMay
    Nov 9 '20 at 7:54

Nowadays, when we "perform" a series of coin toss experiments, then this is often an idealistic thought experiment, or an an experiment with a random number generator on a computer. We are not performing a coin flip experiment in reality.

The behaviour of the coin flip should be such that, not the environment and conditions of the coin flip, but instead the probability of the outcomes, are identical each time. So the term "identical flips" is not clear and should not be used in statistics. It should be "flips with identical probabilities for the outcomes".

It would not be difficult to create a controlled environment were a coin will be flipped in a identical way such that it lands the same everytime, e.g. heads everytime. That is not the idea of 'identical' coin flips (assuming identical refers to identical probability distribution and not identical flipping).

So if you would want to perform a coin flip experiment for real then you would need to have as much 'randomness' as possible in the flipping. But the factor that you wish to keep the same is the probability of the outcomes. Because this is difficult people have been using all sorts of tricks like using numbers/digits from logarithmic tables (which resemble random behaviour) or constructing special machines to manually generate experiments in a random way.

The typical coin flip experiment relates to identical (and also independent) distribution of the outcomes of the coin tosses. Sometimes one considers the coin tosses to be correlated, or biased (the heads and tails probabilitiea for each coin follows some distribution). So, "when we perform a series of coin toss experiments" is a bit ambiguous and not necessarily refering to identical.

  • $\begingroup$ What does then "identicaly distributed" mean? does it mean that data are generated in the same manner? $\endgroup$
    – StoryMay
    Nov 9 '20 at 7:20
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    $\begingroup$ @ChangheeKang it means that the data is generated in such a way that there are fluctuations in the outcomes. The data are not generated in the same manner (at least not with respect to the sources of variation that cause those fluctuations). We consider these fluctuations to be unknown and because of that we apply 'probability' to it, which describes the tendencies of the coin to be either heads or tails. The exact outcomes of the coin are unknown but we do know (more or less) the probability for heads and tails and we assume these probabilities to be the same (identical) each time. $\endgroup$ Nov 9 '20 at 7:27
  • $\begingroup$ We know that there are fluctuations and we present those fluctuations with a probability distribution. When we flip a coin, we don't necessarily think about how each coin is thrown to the air, we consider all the factors that affect how the coin will land as randomness but we assume coins tossed in the same manner. However, we don't really define what the same manner is, that's why we get randomness in the data generation process. $\endgroup$
    – StoryMay
    Nov 9 '20 at 9:08
  • $\begingroup$ We don't define in the 'same manner' because we don't consider the same manner. We only consider the 'same probability of the outcome'. This should be guaranteed by tossing in such a way that the beginning state of the coin toss does not matter, in a reasonably predictable way, for the final outcome. This is done by tossing in such a way that the outcome is due to the mechanics of the coin spinning and tossing, and how it lands is only remotely related to the beginning state, and much influenced by the unknown factors of bouncing on the floor, number of rotations before it lands, etc. $\endgroup$ Nov 9 '20 at 9:11
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    $\begingroup$ @ChangheeKang The process will need to be different in some aspects in order for us to speak about probability. Some things will be different, some things will be the same. Different will be the random parameters/effects that cause the coin to land heads or tails, such that we do not know the outcome. The same will be the parameters/effects that govern the probability on which side the coin will land. This does not need to be described in detail; in whatever way you do it, it does not matter as long as the probability is identical. So in that sense 'the same' is defined. $\endgroup$ Nov 9 '20 at 9:53

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