I am still trying to understand this exercise. I have seen the error in my above post but I cannot edit it so I will post a new one here:

Here is my argument guess after reading the dicussion

here: Because 1,000 coins all share the same \mu and any two flips of one coin are independent from each other, and any two flips of two coins are also independent from each other, and the event a coin is randomly selected is independently from its flip result, so c_{rand} can be treated as a specific coin. Hence the distribution of \nu_{1} and \nu_{rand} is the same and it is binomial distribution. However, \nu_{min} has the different distribution and it is not binomial.

For example:

(*)

while:

Is my argument and calculation correct? I am still confused about the random coin (I think my above argument about random coin is still rather naive).

Is there anything that distinguishes c_rand from c_1?

I see that the result (*): 1 - (1 - 0.5^(10))^(1000) = 0.62357620194... is very close to

giridhar1202's experiemental result, and I also see that (*) is analogous to the coin example that you mentioned in Lecture 02's video. Is my view right?

Thank you very much in advance.