In example 5.4.2 you are sampling from a population $ \prod $ given in example 5.4.1 and $\pi_1,\pi_2,\ldots,\pi_{20}$ are the $20$ members of your sample. Since, there are $20$ members in the population, and the sample you are considering is also of size $20$, the members of every sample will be the same. The difference between samples will be in the order of the sample members.
Note that $X(\pi_i)$ is a Random Variable indicating the fertility of the $i_{th}$ plot in a given sample.
It is equally likely for any of the $20$ plots to be the first, second, or even the tenth plot in a sample. So for a sample that we know nothing about,
$$P(X(\pi_i)\leq x)= F_X(x) \ for \ all \ i=1,2,\ldots,20 $$
where $F_X(x)$ is the CDF of the population $\prod$.
In terms of the first part of the equation, is it given that we don't
know what the value of the first individual of the population set is?
We know the value of the first individual of the population, but that order need not be preserved in a sample from the population.
From my understanding, it's the number of members in the set such that
every member satisfies the condition of being less than "x". If this
is true, I am confused as to how solving for the general population
gives an answer for the probability of the fertility of the FIRST
member being less than "x"? Thanks! –
Your understanding is correct. I can also see where the confusion creeps in. Let's take a simpler example to demonstrate why the two probabilities will always turn out to be the same. Consider the set $A=\{1,2,3\}$ from which we are taking a sample of size $3$. The $3!=6$ possible samples are $S_1=\{1,2,3\}, S_2=\{1,3,2\}, S_3=\{2,1,3\}, S_4=\{2,3,2\}, S_5=\{3,1,2\}, S_6=\{3,2,1\}$. Then, $$P(First\ element\ of\ sample=1)=$$
$$P(First\ element\ of\ sample=2)=$$
$$P(First\ element\ of\ sample=3)=$$
$$\frac{2}{6}=\frac{1}{3}$$
Note that this fraction is the same as $\frac{1}{number\ of\ elements\ in\ population}$. In plain language, the idea being conveyed is that "choosing the first element of the sample" is equivalent to "choosing an arbitrary element from your population". Therefore, solving for the general population gives a valid answer as it perfectly simulates the underlying situation.
PS: I like to think of it as an elegant result of Probability that you get used to with practice.