The correct physical interpretation of Binomial distribution and bernoulli trial in this example

Let $n = $ any large number : say $1000$.

Let $b$ denote the number of blue balls in the bag.

Let $f(b)$ denote the probability of exactly $20\%$ of the $n$ trials succeeding in showing a blue ball, when a ball is selected with replacement from the bag.

Let $W$ denote $\displaystyle \sum_{i = 0}^{100} f(b)$.

Then, the expected number of blue balls in the bag is

$$\frac{\sum_{i=0}^{100} \left[i \times f(i)\right]}{W}.\tag1 $$

$W$ in the denominator serves to normalize the sum of the weights (i.e. the probabilities) associated with each possible number of blue balls.

$\displaystyle f(b) = \binom{1000}{200} \times \left[\frac{b}{100}\right]^{(200)} \times \left[\frac{100 - b}{100}\right]^{(800)}.$