I am hung up on this question too.
As I interpret it, a "bin" in this question corresponds to a particular way for defining a random experiment. In any case, we flip 1000 coins 10 times each and:
1) the random experiment for coin 1 consists of taking the 10 flips for that coin and averaging the number of heads, reporting that average as the sample frequency.
2) For the random coin, the random experiment consists in taking a random coin (11000) and averaging out the number of heads in the 10 flips for that coin.
3) for the "minimum number of heads coin", we flip 1000 coins 10 times each as before, but take the minimum frequency of heads obtained among the 1000 coins as the reported sample frequency.
Each of these 3 situations constitutes a random experiment because it can be replicated any number of times. Each of the three scenarios has an expected frequency. The first two have an expected frequency (mu) of 0.5, the third has a different expected frequency but it can be calculated analytically (the probability distribution for the possible sample frequencies 0, 0.1, ..., 0.9, 1 can be determined for the "minimum heads" coin). The sample frequencies converge to the respective expected frequencies for the three scenarios, according to the law of large numbers. (No surprise)...
So back to this:
Quote:
If you fix any hypothesis then run the experiment, the probability will always be bounded by that term.

Does fixing the hypothesis mean fixing the coins, or does it mean being able to fix a random experiment (procedure) by which we obtain the sample frequencies?