Probability Coin and Dice fallacy? In a coin experiment, one may toss two coins simultaneously, or one after the other. Does it make a difference if the coins are indistinguishable or distinct?
For me, if two coins are tossed simultaneously, then the sample spaces are:
Indistinguishable coins:
$$
\{\{H, H\}, \{H, T\}, \{T, T\}\}.
$$
Distinct coins:
$$
\{HH, HT, TH, TT\}
$$
Is this correct? And if a question is given like "If three coins are tossed simultaneously, what is the probability that we obtain two heads and one tail?". In this question, what is the case we should take, as nothing is mentioned about the coins?
 A: Suppose you had two coins, lightly painted blue and green so that they are clearly distinct, and use a mechanical device to toss them as closely as possible in time and space, wait a moment so the result can be recorded, then retrieve them for the next toss.   Thus determining the probabilities of the results through practical.
Next, without disturbing the setup in any way, we place a box of red-tinted glass over the device so that any observer cannot tell which coin is which—their distinction is hidden by colour filtration—, and then repeat the experiment again.
Would you anticipate that the fact that any observer cannot distinguish the coins during the second experiment would somehow affect the probability of the outcomes?   Or would you anticipate that being able or unable to distinguish the coins would have no such affect at all?   (Beyond, that is, being unable to distinguish between $\rm\color{green}{H}\color{blue}{T}$ and $\rm\color{blue}{H}\color{green}{T}$ when they both appear to be $\rm{HT}$.)
A: The sample space is {HH},{HT},{TT} but the probability of sample {HT} is 1/2 since we can get heads on the first coin and tails on the second or vice versa, regardless of whether the two coins are tossed simultaneously or with infinitesimal time difference.
