Ian Segment 1
To Think About-2
As in the case of "", we can first represent as a binary number (maybe infinite number of digits), and then,
For k-th coin flip, encoding "Head" as 1, "Tail" as 0,
if we get a number with > k-th digit of , then we answer "no".
If < k-th digit of , then we answer "yes".
If they are equal, we continue the experiment, until one of the above case happens.
In expectation, we need only 2 times of flips, since every trial we have only chance to get into the "re-try" case. The number of trials we need follows a geometrical distribution with 0.5 success probability, which has E[#trails]=2.
To Think About-3
We can model the system of "fishes" as a Markov Chain where the state is represented as (#minnows,#trout), which has a absorbing state (3,0) that cannot transit to any other states anymore.
We can construct a transition matrix A using the probability defined in the problem, and initial probability distribution x0 that has state (3,2) with probability 1.
The distribution of states after N trials can be represented as , the probability N-th fish is a trout can be represented as
, where x[i,j] means the probability entry in x of the state (i,j).